A Single-arm, Multicenter, Open-label, Phase I/II Trial of Allo-QuadCAR01-T, an Allogeneic CAR-T-cell Therapy Targeting CD19 and CD20, for the Treatment of Relapsed or Refractory B-cell Malignancies

Inclusion Criteria: * Adults 18 years or older. * Diagnosed with relapsed or refractory B-cell non-Hodgkin lymphoma (B-NHL) or chronic lymphocytic leukemia (CLL). * Must have received at least 2 prior lines of therapy. * ECOG performance status 0-1 (able to carry out daily activities). * Adequate organ function (heart, liver, kidneys). * HLA B/C match with donor cells. * No active uncontrolled infections. Exclusion Criteria: * Active CNS involvement (including PCNSL) in dose escalation cohorts; may be allowed in later cohorts with Sponsor approval. * Prior CAR-T within 3 months of screening, or ≥Grade 3 ICAHT from prior CAR-T. * Autologous stem cell transplant within 3 months. * Prior allogeneic stem cell transplant or solid organ transplant. * Prior therapy with dual CD19/CD20 CAR-T. * Severe hypersensitivity to trial agents or similar compounds. * History of GvHD or post-transplant lymphoproliferative disorder. * Presence of La/SS-B autoantibodies or related autoimmune diseases. * Other malignancy that may interfere with trial, except: * Curatively treated basal/squamous skin cancer or cervical carcinoma in situ * Low-grade, early-stage prostate cancer (Gleason ≤6, Stage 1-2) with no therapy needed * Adjuvant endocrine therapy for non-metastatic breast cancer (≥2 years) * Any other curatively treated malignancy in remission ≥2 years * Active viral infection within 1 week of screening, or serious bacterial/fungal infection. * Hemorrhagic cystitis. * Active neuro-autoimmune disease (e.g., MS, Guillain-Barré, ALS). * Active or residual HBV, HCV, or syphilis. * Active HIV. History of HIV may be eligible with Sponsor approval if: * Neurological disorders within 6 months (e.g., stroke, dementia, Parkinson's, cerebellar disease, CNS autoimmune disease). * Significant cardiac disease within 6 months (e.g., MI, stent, unstable angina). * Primary immunodeficiency or autoimmune disease requiring systemic treatment within 1 year (unless stable and Sponsor-approved). * Unresolved ≥Grade 2 non-hematologic toxicity from prior therapy (except neuropathy up to Grade 2). * Systemic immunosuppression within 28 days. * Last systemic lymphoma/CLL therapy (standard or investigational) within 28 days or 5 half-lives. * Major surgery within 14 days. * Local radiation within 28 days. * Live vaccination within 28 days. * Pregnant or breastfeeding.