This is a Phase I, prospective, dose-finding study to evaluate the safety, persistence, and clinical activity of GF-CART01 in subjects aged 18-70 with relapsed or refractory (R/R) B-cell hematological malignancies and failure of two-line or more standard chemotherapies or auto-hematopoietic stem cell transplantation (HSCT).This study is a traditional 3+3 dose-escalation design to observe dose-limiting toxicity (DLT), establish the maximum tolerated dose(MTD)/recommended phase 2 doses (RP2D), and preliminary efficacy of GF-CART01. RP2D may equal to or lower than MTD

Inclusion Criteria:

1. Subjects must be of age ≥ 18 years and ≤ 70 years
2. Subjects or their legal guardians must volunteer to participate in the study and sign the informed consent
3. Histologically confirmed diagnosis of Diffuse Large B-Cell Lymphoma - Not Otherwise Specified (DLBCL-NOS), follicular lymphoma (FL), Primary Mediastinal Large B-cell Lymphoma (PMBCL), or High-Grade B-Cell Lymphoma (HGBCL) per the world health organization (WHO) Classification Criteria for Lymphoma (2022)
4. Tumor cell surface expression of CD19 (+) and/or CD20 (+) by flow cytometry or immunohistochemistry staining
5. Relapsed, progressive or refractory disease (defined as have not achieved a complete response) after ≥ two lines of systemic therapy, including anti-CD20 antibody and anthracycline and/or Relapsed, progressive or refractory disease ( defined as have not achieved a complete response) after auto-HSCT
6. Subjects have any accessible PET-positive lesion or measurable CT-positive lesion per Lugano 2014 criteria
7. Adequate hematologic function: absolute neutrophil count (ANC) > 1,000/μL, absolute lymphocyte count (ALC) > 300/μL, platelet count ≥ 75,000/μL, hemoglobin ≥ 8.0 g/dL
8. Adequate hepatic function: alanine aminotransferase (ALT) ≤ 5 times upper limit of normal (ULN), aspartate transaminase (AST) ≤ 5 times ULN, total bilirubin ≤ 1.5 times ULN
9. Adequate renal function: blood estimated glomerular filtration rate (eGFR) ≥ 60 mL/min/1.73m2 (calculated by Modification of Diet in Renal Disease (MDRD) equation)
10. Adequate cardiac function: echocardiogram or multigated blood pool analysis (MUGA) shows left ventricular ejection fraction (LVEF) ≥ 50%; and no clinically significant electrocardiogram (ECG) findings
11. Adequate pulmonary function: no active infection in the lungs, blood oxygen saturation in indoor air ≥ 92%
12. No clinically significant pleural effusion determined by the investigators
13. Estimated survival time ≥ 3 months
14. Eastern Cooperative Oncology Group (ECOG) performance status 0 to 2
15. Willingness and ability to comply with protocol-stated requirements, instructions, and restrictions in the investigator's judgement

Exclusion Criteria:

1. Previously treated with any CAR T cell product or allogenic hematopoietic stem cell transplant (HSCT)
2. Known or suspected allergy, hypersensitivity, or intolerance to any ingredients of the investigational product (IP)
3. Known medical history or possible risk for taking contrast agent(s) for positron emission tomography (PET) and/or computed tomography (CT) scan
4. Received any other investigational product, cell therapy, or gene therapy within 12 weeks prior to the leukapheresis
5. Received any tyrosine kinase inhibitor within 2 weeks prior to the leukapheresis
6. Received any systemic steroid, immunotherapy (such as immune checkpoint inhibitors, T- cell transfer therapies, monoclonal antibodies), or chemotherapy within 4 weeks prior to the leukapheresis
7. Received any live vaccine from 2 weeks prior to the leukapheresis
8. Subjects with human immunodeficiency virus (HIV), syphilis, Hepatitis B or C infection: HIV-1 and HIV-2 antibody positive, syphilis antibody positive, both hepatitis B virus (HBV) DNA and hepatitis B core (HBc) antibody positive, or hepatitis C virus (HCV) antibody positive
9. Subjects with atrial or ventricular involvement by B-cell malignancies
10. Subjects with tumor mass requiring urgent treatment within 8 weeks prior to the leukapheresis, such as ileus, tumor lysis syndrome, or vascular compression
11. Subjects with severe disease or other uncontrolled diseases, such as coronary heart disease, angina pectoris, myocardial infarction, arrhythmia (urgent intervention indicated, life- threatening consequences, or hemodynamic compromise), cardiac angioplasty or stenting, unstable angina, cerebral thrombosis, cerebral hemorrhage, hypertension (systolic blood pressure ≥ 140 mmHg and/or diastolic blood pressure ≥ 90 mmHg)
12. Unstable pulmonary embolism, deep venous embolism, or other major arterial/venous thromboembolism events that occurred within 6 weeks prior to the leukapheresis. If subjects receive anticoagulant therapy, the treatment dose and frequency must be stable for more than 14 weeks prior to the leukapheresis
13. History of craniocerebral trauma, disturbance of consciousness, epilepsy, cerebrovascular ischemia, cerebrovascular hemorrhagic diseases, dementia, cerebellar disease, or any autoimmune disease with central nervous system (CNS) involvement
14. Female subject of childbearing potential who: a. Has positive pregnancy test result; or b. Is lactating
15. Female subjects of childbearing potential, or male subject with female spouse/partner of childbearing potential, who refuses to adopt at least one form of birth control from the date of signing informed consent to 12 months after GF-CART01 infusion. Acceptable forms of birth control include: a. Established use of oral, injected or implanted hormonal methods of contraception b. Placement of an intrauterine device (IUD) or intrauterine system (IUS) c. Barrier methods of contraception: condom or occlusive cap (diaphragm or cervical/vault caps)
16. Any following situations that the investigators believe are not suitable for this trial and/or may increase the risk for subjects or interfere with the results of the study a. With severe active infections (except simple urinary tract infection and bacterial pharyngitis) b. With active central nervous system involvement by lymphoma, malignant cells in cerebrospinal fluid c. History of brain metastasis d. With uncontrolled malignancies e. Any toxicities due to prior therapy f. With any uncontrolled illness or a history of any illness