Bispecific CAR T Cells for B-cell Malignancies (BaseCAR-01 Trial)

Inclusion Criteria: * Age ≥ 18 years * Diagnosis of B-cell NHL or B-ALL with relapsed, refractory disease and no available standard therapeutic options (including commercially accessible CAR T products), including: * Acute B-lymphoblastic leukaemia * Burkitt lymphoma * Primary CNS lymphoma * DLBCL or high-grade lymphoma of any subtype * Primary mediastinal B cell lymphoma (including grey zone lymphoma) * Mantle Cell lymphoma * Low-grade B-cell NHLs: Follicular lymphoma, chronic lymphocytic leukaemia (CLL)/ small lymphocytic lymphoma (SLL), marginal zone lymphoma, hairy cell leukaemia, splenic B-cell lymphoma/leukaemia with prominent nucleoli, and lymphoplasmacytic lymphoma * CD19 and/or CD20 positive disease on most recent evaluation (by immunohistochemistry or flow cytometry) * ECOG clinical performance status ≤2 * Able to provide written informed consent. * Adequate organ function and bone marrow reserve, unless clearly caused by lymphoma and considered reversible, defined as: * Adequate hepatic function: Aspartate Aminotransferase (AST) and Alanine Aminotransferase (ALT) ≤3.0 × (ULN) and serum bilirubin ≤2.0 × ULN (except in congenital hyperbilirubinemia, such as Gilbert syndrome, where direct bilirubin ≤3.0 × ULN is allowed) * Adequate renal function: creatinine clearance ≥30 mL/min/1.73 m2 * Adequate pulmonary function: Forced Expiratory Volume in 1 second (FEV1) ≥50% (with adequate compliance) and pulse oxygenation \> 91% with room air. * Adequate cardiac function: Left Ventricular Ejection Fraction (LVEF) ≥ 40%, and no clinically significant arrhythmia * Adequate bone marrow reserve (Hemoglobin ≥80 g/L (with or without recombinant erythropoietin or red blood cell transfusions), Platelets ≥ 50×10\^9/L (with or without platelet transfusions), Absolute Neutrophil Count (ANC) 1.0 ×10\^9/L (prior growth factor support is permitted but must be without support in the 7 days before the laboratory test), Absolute Lymphocyte Count ≥0.3 ×10\^9/L) * Willingness to practice highly effective methods of birth control, and, in females of childbearing potential, negative urine or serum pregnancy test before study inclusion, lymphapheresis, and lymphodepleting chemotherapy. Exclusion Criteria: * Requirement for systemic corticosteroids, i.e. ≥20 mg of prednisone or equivalent daily. Other immunosuppressive drugs * Any organ failure, respectively not meeting the inclusion criteria of adequate organ function, or active, BKuncontrolled autoimmune disease. * Uncontrolled coronary artery disease or uncontrolled arrhythmias * Stroke within the previous 6 months, a history of neurodegenerative disorder or overt clinical evidence of dementia or altered mental status. * Seizure within 6 months of signing the ICF unless related to the primary disease (e.g. CNS lymphoma). * Active secondary malignancy that progressed or required treatment in the last 24 months, other than basal or squamous cell carcinomas of the skin. Further allowed exceptions are: Non-muscle-invasive bladder cancer, non-invasive cervical cancer, or other malignancy that is considered cured or to have a minimal risk of recurrence (e.g. a history of localized prostate or localized and treated breast cancer). * Uncontrolled active bacterial, fungal, or viral infections, particularly active hepatitis B, hepatitis C, or HIV infection. * Contraindications, known life-threatening allergies, hypersensitivity, or intolerance to any of the study treatments, including previous severe reactions to dimethyl-sulfoxide * Cytotoxic chemotherapy within 14 days before apheresis collection for CAR-T cells, respectively 12 weeks for Bendamustin and Fludarabine, and 6 months for Alemtuzumab and ATG. * Cytotoxic chemotherapy (except for lymphodepletion) within 14 days of CAR-T cell infusion. * Patients who have undergone allogeneic hematopoietic stem cell transplantation less than 12 weeks ago, have evidence of active graft-versus-host-disease (GVHD) of any grade, or are currently on immunosuppression. * Previous CAR-T cell therapy within 12 weeks of planned CAR-T cell infusion. * Investigational treatments within other trials ≤ 4 weeks before enrollment. * Lack of safe contraception; Women who are pregnant or breastfeeding; and men who plan to father a child while enrolled in this study within 1 year of receiving bispecific anti-CD20, anti-CD19 CAR T cells.