This study is testing Allo-QuadCAR01-T, a new off-the-shelf CAR-T therapy for people with hard-to-treat B-cell cancers. Unlike current CAR-T treatments that use a patient's own cells, this therapy uses donor cells that are ready to use, which can save time and reduce costs. It targets two proteins, CD19 and CD20, to lower the chance of relapse and uses gene editing to make it safer. The trial has three parts: first to find a safe dose, then to confirm it, and finally to test how well it works in patients with diffuse large B-cell lymphoma (DLBCL). Patients will get one infusion after chemotherapy to prepare their body. The main goal is to check safety and see how many patients have a complete response by Week 13. About 160 patients will take part, and researchers will follow them for up to 15 years.

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.