This is an open-label Phase 1b/2 trial of lenvatinib (E7080) plus pembrolizumab in participants with selected solid tumors. Phase 1b will determine and confirm the maximum tolerated dose (MTD) for lenvatinib in combination with 200 milligrams (mg) (intravenous [IV], every 3 weeks [Q3W]) pembrolizumab in participants with selected solid tumors (i.e. non-small cell lung cancer, renal cell carcinoma, endometrial carcinoma, urothelial carcinoma, squamous cell carcinoma of the head and neck, melanoma or leiomyosarcoma). Phase 2 (Expansion) will evaluate the safety and efficacy of the combination in 7 cohorts at the MTD from Phase 1b (lenvatinib 20 mg/day orally + pembrolizumab 200 mg Q3W, IV).

Inclusion Criteria:

1. Phase 1b: Histologically and/or cytologically confirmed metastatic selected solid tumor types that have progressed after treatment with approved therapies or for which there are no standard effective therapies available. If nivolumab or pembrolizumab is an approved therapy for the participant's tumor type, but the participant has not been treated with it, the Investigator may enroll the participant in this study.

   Phase 2: Histologically and/or cytologically confirmed metastatic selected solid tumor types with 0-2 prior lines of systemic therapy. If previously treated, participant has progressed after previous treatment. For the non-small cell lung cancer (NSCLC) and melanoma cohorts, participants must have progressed on or after prior treatment with one anti-programmed cell death protein 1 (anti-PD-1), anti-PD-1 ligand 1 (anti-PD-L1), or anti-PD-1 ligand 2 (anti-PD-L2) agent. For the renal cell carcinoma (RCC) cohort, participants must have progressed on treatment with an anti- programmed death receptor-1 /programmed death receptor-ligand 1 monoclonal antibody (anti-PD-1/PD-L1 mAb) administered either as monotherapy, or in combination with other checkpoint inhibitors or other therapies, the regimen with an anti-PD-1/PD-L1 mAb must be the most recent therapy. Selected tumor types of both phases: NSCLC, predominantly clear cell renal cell carcinoma, endometrial carcinoma, urothelial carcinoma, squamous cell carcinoma of the head and neck, or melanoma (excluding uveal melanoma)

2. Life expectancy of 12 weeks or more

3. Phase 2: Measurable disease meeting the following criteria:

   1. At least 1 lesion of greater than or equal to 10 mm in the longest diameter for a non-lymph node or greater than or equal to 15 mm in the short-axis diameter for a lymph node that is serially measurable according to irRECIST (immune-related Response Evaluation Criteria in Solid Tumors) using computerized tomography/magnetic resonance imaging (CT/MRI)
   2. Lesions that have had external beam radiotherapy (EBRT) or loco-regional therapies such as radiofrequency (RF) ablation must show subsequent evidence of substantial size increase to be deemed a target lesion

4. Participants must have an Eastern Cooperative Oncology Group (ECOG) Performance Status of 0 to 1

5. Adequately controlled blood pressure (BP) with or without antihypertensive medications, defined as BP less than or equal to 150/90 mmHg at screening and no change in antihypertensive medications within 1 week prior to the Cycle 1 Day 1

6. Adequate renal function defined as creatinine less than or equal to 1.5*ULN (upper limit of normal) or calculated creatinine clearance greater than or equal to 40 mL/min per the Cockcroft and Gault formula with creatinine levels greater than 1.5*ULN

7. Adequate bone marrow function:

   1. Absolute neutrophil count (ANC) greater than or equal to 1500/mm3 (greater than or equal to 1.5 X 103/uL)
   2. Platelets greater than or equal to 100,000/mm3 (greater than or equal to 100 X 109/L)
   3. Hemoglobin greater than or equal to 9.0 g/dL

8. Adequate blood coagulation function as evidenced by an International Normalized Ratio (INR) less than or equal to 1.5

9. Adequate liver function as evidenced by bilirubin less than or equal to 1.5 times the ULN and alkaline phosphatase (ALP), alanine aminotransferase (ALT), and aspartate aminotransferase (AST) less than or equal to 3*ULN (in the case of liver metastases less than or equal to 5*ULN). In case ALP is greater than 3 X ULN (in the absence of liver metastases) or greater than 5 X ULN (in the presence of liver metastases) AND the participant also is known to have bone metastases, the liver specific ALP must be separated from the total and used to assess the liver function instead of the total ALP

10. Males or females age greater than or equal to 18 years at the time of informed consent

11. Participants with known brain metastases will be eligible if they have completed the primary brain therapy (such as whole brain radiotherapy, stereotactic radiosurgery or complete surgical resection) and if they have remained clinically stable, asymptomatic and off of steroids for at least 28 days before starting study treatment.

12. All females must have a negative serum or urine pregnancy test (minimum sensitivity 25 IU/L or equivalent units of beta-human chorionic gonadotropin [β-hCG]) at the Screening Visit and the Baseline Visit. A pregnancy test needs to be performed within 72 hours of the first dose of study drug. Females of childbearing potential must agree to use a highly effective method of contraception for the entire study period and for 120 days after study discontinuation, ie

    • total abstinence (if it is their preferred and usual lifestyle)
    • an intrauterine device (IUD) or hormone-releasing system (IUS)
    • a contraceptive implant
    • an oral contraceptive** (with additional barrier method) OR
    • have a vasectomized partner with confirmed azoospermia.

    NOTES:

    • All females will be considered to be of childbearing potential unless they are postmenopausal (amenorrheic for at least 12 consecutive months, in the appropriate age group, and without other known or suspected cause) or have been sterilized surgically (ie, bilateral tubal ligation, total hysterectomy, or bilateral oophorectomy, all with surgery at least 1 month before dosing).
    • Must be on a stable dose of the same oral hormonal contraceptive product for at least 4 weeks before dosing with study drug and for the duration of the study

13. Male participants who are partners of women of childbearing potential must use a condom + spermicide and their female partners if of childbearing potential must use a highly effective method of contraception (see methods described in Inclusion Criterion #12) beginning at least 1 menstrual cycle prior to starting study drug(s), throughout the entire study period, and for 120 days after the last dose of study drug, unless the male participants are totally sexually abstinent or have undergone a successful vasectomy with confirmed azoospermia or unless the female partners have been sterilized surgically or are otherwise proven sterile.

14. Voluntary agreement to provide written informed consent and the willingness and ability to comply with all aspects of the protocol

15. Archival tumor tissue or a newly obtained biopsy must be available prior to the first dose of study drug for biomarker analysis. In the case archival tissue cannot be provided, participants with inaccessible tumors for biopsy specimens can be enrolled without a biopsy upon consultation and agreement by the sponsor Note: In case of submitting unstained cut slides, freshly cut slides should be submitted to the testing laboratory within 14 days from when the slides are cut.

Exclusion Criteria:

1. Prior anticancer treatment within 28 days (or 5 times the half-life time, whichever is shorter) or any investigational agent within 30 days prior to the first dose of study drugs. All acute toxicities related to prior treatments must be resolved to Grade less than or equal to 1
2. Participants must have recovered adequately from any toxicity and/or complications from major surgery prior to starting therapy
3. Participants having greater than 1+ proteinuria on urinalysis will undergo 24-h urine collection for quantitative assessment of proteinuria. Participants with urine protein greater than or equal to 1 g/24-hour will be ineligible.
4. Gastrointestinal malabsorption, gastrointestinal anastomosis, or any other condition that might affect the absorption of lenvatinib
5. New York Heart Association congestive heart failure of grade II or above, unstable angina, myocardial infarction within the past 6 months, or serious cardiac arrhythmia associated with significant cardiovascular impairment within the past 6 months
6. Prolongation of corrected QT (QTc) interval to greater than 480 msec
7. Active hemoptysis (bright red blood of at least 0.5 teaspoon) within 3 weeks prior to the first dose of study drug
8. Active infection (any infection requiring systemic treatment)
9. Participant is known to be positive for Human Immunodeficiency Virus (HIV), Hepatitis B, or Hepatitis C
10. Serious nonhealing wound, ulcer, or bone fracture
11. Known intolerance to either of the study drugs (or any of the excipients)
12. History of organ allograft (Participant has had an allogenic tissue/solid organ transplant)
13. Biologic response modifiers (eg, granulocyte colony-stimulating factor) within 4 weeks before study entry. Chronic erythropoietin therapy is permitted provided that no dose adjustments were made within 2 months before first dose of study treatment
14. Any medical or other condition which, in the opinion of the investigator, would preclude participation in a clinical trial
15. Females who are pregnant or breastfeeding
16. Excluding the primary tumor leading to enrollment in this study, any other active malignancy (except for definitively treated melanoma in-situ, basal or squamous cell carcinoma of the skin, or carcinoma in-situ of the bladder or cervix) within the past 24 months
17. Prior treatment with lenvatinib or any PD-1, anti-PD-L1, or anti-PD-L2 agent, excluding melanoma and NSCLC where prior treatment with one PD-1, anti-PD-L1, or anti-PD-L2 agent is allowed, and excluding RCC where prior treatment with one regimen containing an anti-PD-1/PD-L1 mAb is required.
18. Has a diagnosis of immunodeficiency or is receiving systemic steroid therapy or any other form of immunosuppressive therapy within 7 days prior to the first dose of study treatment. The use of physiologic doses of corticosteroids (up to 7.5mg/d of prednisone or equivalent) may be approved after consultation with the sponsor.
19. No active autoimmune disease that has required systemic treatment in past 2 years (ie, with use of disease modifying agents, corticosteroids or immunosuppressive drugs). Replacement therapy (eg, thyroxine, insulin, or physiologic corticosteroid replacement therapy for adrenal or pituitary insufficiency, etc.) is not considered a form of systemic treatment.
20. Has a history of (non-infectious) pneumonitis that required steroids or current pneumonitis
21. Has received a live-virus vaccination within 30 days of planned treatment start. Seasonal flu vaccines that do not contain live virus are permitted.