This study will recruit patients with the following cancer indications: ovarian cancer, squamous non-small cell lung cancer, synovial sarcoma and head and neck cancer, with inoperable locally advanced or metastatic solid tumours. Currently, these patients have a poor prognosis and a relatively short overall survival. There is a lack of meaningful, effective therapies available that improve the outcome for these patients. The treatment being investigated in this study is ZIMA4-1, an allogeneic cell therapy product. This is the first time ZI-MA4-1 will be administered to humans. The study is planned to consist of two parts (A and B). Part A includes up to four dose escalation cohorts and aims to identify the maximum tolerated dose of ZI-MA4-1 and give insight into the recommended Phase 2 dose (RP2D). Part B consists of an expansion cohort and is designed to further evaluate the RP2D identified in Part A across one or more indications. The study procedures and eligibility criteria will be the same for participants in Parts A and B, except for the dose level of ZI-MA4-1.

Inclusion Criteria:

• HLA-A*02:01 positive
• Tumour(s) show expression of the MAGE-A4 protein above a defined threshold
• Histopathological or cytological diagnosis of inoperable Locally Advanced or Metastatic malignant disease: ovarian cancer, squamous non-small cell lung cancer (NSCLC), synovial sarcoma or head and neck cancer.
• No approved therapy with demonstrated clinical benefit is indicated or available to treat the patient, or the patient is intolerant of or has refused standard of care therapy.
• Documented imaging confirmed disease progression while on or within 6 months after the end of the most recent therapy.
• Participant must have received ≥2 prior lines of cancer therapy except for patient with synovial sarcoma for whom ≥1 prior lines of cancer therapy.
• Measurable disease according to RECIST v1.1 criteria.
• ECOG performance status of 0 or 1 with no deterioration over the previous 2 weeks, and an anticipated life expectancy of >3 months
• Female participants are eligible to participate if they are not pregnant or breastfeeding. Woman of childbearing potential must have negative pregnancy test and agree to use an effective contraceptive method.

Other protocol defined inclusion criteria could apply.

Exclusion Criteria:

• Patients have received any prior cellular or gene therapy.
• Receiving experimental investigational products within 4 weeks of lymphodepletion.
• Recent therapies (within up to 4 weeks prior to lymphodepletion) including biologic agents (such as monoclonal antibodies), anti-cancer immunotherapy (such as monoclonal antibodies against PD-1 receptor or ligand).
• Residual toxicities ≥2 CTCAE grade due to prior therapy, that in the opinion of the investigator may interfere with study conduct.
• Any other active malignancy besides the tumour under study within 3 years prior to screening except for in situ removal of basal cell carcinoma or adequately treated cervix carcinoma in-situ.
• Active or documented history of autoimmune disease or any other diseases requiring immunosuppressive therapy or corticosteroid therapy. Physiological replacement, topical, and inhaled steroids are permitted.
• Significant CNS disorders.
• Myocardial infarction, cardiac angioplasty or stenting, cardiac arrhythmia requiring medication, unstable angina, New York Heart Association Class II or greater congestive heart failure, cardiac atrial or ventricular lymphoma involvement, or other clinically significant cardiac disease within 6 months of enrolment.
• Active fungal, bacterial viral, or other infection requiring intravenous antibiotic, antifungal, or antiviral medication within 7 days prior to lymphodepletion.
• Received or planned to receive a live vaccine ≤6 weeks before the planned start date of lymphodepletion.
• Severe or uncontrolled medical condition (e.g., severe chronic obstructive pulmonary disease, interstitial lung disease , severe Parkinson's disease, active inflammatory bowel disease) or psychiatric condition, which in the opinion of the investigator would interfere with study activities.
• Active bleeding diatheses, including but not limited to therapeutic anticoagulation, and treatment with major surgery within 28 days before lymphodepletion (minor surgical procedures such as lymph node biopsy/excision or catheter placement are permitted).
• Patients have significant immunosuppression
• Known significant hepatic or biliary abnormalities. Active infection with hepatitis B, hepatitis C.
• Any medical, psychological, or social condition, drug or alcohol abuse that would make it difficult for the patient to participate in the study and comply with the study procedures, restrictions, and requirements.
• History of allergic reactions to compounds chemically or biologically similar to cyclophosphamide, fludarabine or other agents used in the study
• QTc > 450 msec for male participants or > 470 msec for female participants
• Medical conditions, such as anti-coagulation, which is not suitable for reversal which, at the opinion of the investigator, preclude or make the patient a poor candidate for biopsy.
• Personal history of allergies or intolerance to local anaesthetic.
• Recent treatment with immunosuppressive agents
• Residual toxicities ≥2 Common Terminology Criteria for Adverse Events (CTCAE) grade due to prior therapy, that in the opinion of the investigator may interfere with study conduct Other protocol defined exclusion criteria could apply.