Inclusion Criteria:

1\. Male or female 18-75 years of age. 2. Patient with:

a. NHL: Histologically confirmed B cell NHL (including but not limited to diffuse large B cell lymphoma (DLBCL), follicular lymphoma, MCL, marginal zone lymphoma (MZL)) i. Relapsed after 2 or more lines of therapy, or ii. Have disease refractory to prior chemotherapy (defined as progressive disease or stable disease lasting ≤ 6 months, as best response to most recent chemotherapy regimen; or disease progression, or recurrence ≤ 12 months after prior autologous stem cell transplantation (ASCT), and iii. Measurable disease per Lugano Revised Response Criteria for Malignant Lymphoma

or

b. CLL: histologically confirmed CLL i. Relapsed after 2 or more lines of therapy ii. Previous therapies prescribed must have included a Bruton's tyrosine kinase (BTK) inhibitor and a B-cell lymphoma 2 (BCL2) inhibitor, iii. Measurable disease and active disease: Active disease as defined by the iwCLL criteria, meeting at least one of the following criteria:

1. Evidence of progressive marrow failure as manifested by the development of, or worsening of, anemia and/or thrombocytopenia. Cutoff levels of Hb \<10 g/dL or platelet counts \<100 × 10\^9/L are generally regarded as indication for treatment.
2. Massive (i.e., ≥6 cm below the left costal margin) or progressive or symptomatic splenomegaly.
3. Massive nodes (i.e., ≥10 cm in longest diameter) or progressive or symptomatic lymphadenopathy.
4. Symptomatic or functional extranodal involvement (e.g., skin, kidney, lung, spine, etc.).
5. Disease-related symptoms as defined by any of the following:

1. Unintentional weight loss ≥10% within the previous 6 months.
2. Significant fatigue (i.e., Eastern Cooperative Oncology Group (ECOG) performance scale 2 or worse; cannot work or unable to perform usual activities).
3. Fevers ≥ 38.0°C for 2 or more weeks without evidence of infection.
4. Night sweats for ≥1 month without evidence of infection.

or

c. HCL: histologically confirmed HCL i. Relapsed after at least one line of therapy, which must have included a purine nucleoside (eg. fludarabine, cladribine or pentostatin) and moxetumomab pasudotox.

ii. Need for treatment as evidenced by any one of the following: Absolute Neutrophil Count (ANC) \<1 × 10\^3/mcL, Hb \<10g/dL, platelet count \<100 × 10\^3/mcL, leukemia cell count \>5 × 10\^3/mcL, symptomatic splenomegaly, or enlarging HCL mass \>2 cm in short axis.

or

d. MM: histologically confirmed MM i. Relapsed or refractory after 3 or more lines of therapy including an immunomodulatory agent, a proteasome inhibitor and an anti-CD38 antibody.

ii. Measurable disease per IMWG uniform response criteria 3. Adequate organ function as defined by:

1. Creatinine clearance more than or equal to 45 ml/min calculated by the Cockcroft - Gault formula
2. Subjects must have adequate cardiac function as defined as left ventricular ejection fraction ≥ 45% on the most recent echocardiogram and no clinically significant arrhythmias, pericardial effusion, valvular, or ischemic heart disease.
3. Adequate pulmonary function with pulse oximetry ≥ 92% on room air.
4. Total Bilirubin \< 1.5× the institutional upper limit of normal (\<2.5× if caused by the baseline cancer and in patients with Gilbert's syndrome).
5. Alanine aminotransferase (ALT (Serum Glutamic-Pyruvic Transaminase (SGPT))) and Aspartate Aminotransferase (AST (Serum Glutamic-Oxaloacetic Transaminase (SGOT) \< 2.5× the institutional upper limit of normal.

4\. Subjects (or legal guardians) must have the ability to understand and the willingness to sign a written informed consent document.

5\. For women of childbearing potential: agreement to remain abstinent (refrain from heterosexual intercourse) or use a contraceptive method with a failure rate of \< 1% per year during the treatment period and for at least 1 year after CAR-T cell infusion.

A woman is considered to be of childbearing potential if she is postmenarcheal, has not reached a postmenopausal state (\< 12 continuous months of amenorrhea with no identified cause other than menopause), and has not undergone surgical sterilization (removal of ovaries and/or uterus).

Examples of contraceptive methods with a failure rate of \< 1% per year include bilateral tubal ligation, male sterilization, hormonal contraceptives that inhibit ovulation, hormone-releasing intrauterine devices, and copper intrauterine devices.

The reliability of sexual abstinence should be evaluated in relation to the duration

of the clinical trial and the preferred and usual lifestyle of the patient. Periodic abstinence (e.g., calendar, ovulation, symptothermal, or postovulation methods) and withdrawal are not acceptable methods of contraception.

6\. For men: agreement to remain abstinent (refrain from heterosexual intercourse) or use contraceptive measures, and agreement to refrain from donating sperm, as defined below: With female partners of childbearing potential, men must remain abstinent or use a condom plus an additional contraceptive method that together result in a failure rate of \< 1% per year during the treatment period and for at least 1 year after CAR-T cell infusion. Men must refrain from donating sperm during this same period. With pregnant female partners, men must remain abstinent or use a condom during the treatment period and for at least 1 year after CAR-T cell infusion to avoid potential embryonal or fetal exposure.

The reliability of sexual abstinence should be evaluated in relation to the duration of the clinical trial and the preferred and usual lifestyle of the patient. Periodic abstinence (e.g., calendar, ovulation, symptothermal, or postovulation methods) and withdrawal are not acceptable methods of contraception.

7\. Body weight of at least 55kg for patients treated at dose level 3 (450 × million BAFF+ CAR cells) and at least 37kg for all other dose levels.

Exclusion Criteria:

1. Second active (i.e., currently requires antineoplastic therapy) non-B cell lineage malignancy, other than non-melanoma skin cancer or carcinoma in situ (e.g. cervix, bladder, breast).
2. Renal failure requiring regular dialysis.
3. Uncontrolled pulmonary disease or infection.
4. Cardiovascular disorders including symptomatic congestive heart failure, unstable angina pectoris, clinically significant cardiac arrhythmias, myocardial infarction or stroke (including transient ischemic attack, or other ischemic event) within 6 months prior to registration.
5. Active infection requiring systemic treatment.
6. HIV seropositive with a history of acquired immunodeficiency syndrome (AIDS)-defining opportunistic infections within 12 months of enrollment, or has not been on an established antiretroviral therapy (ART) for at least four weeks with an HIV viral load less than 400 copies/mL.
7. Pregnant or breastfeeding women are excluded from this study (breastfeeding should be discontinued), because there is an unknown, but potential risk for adverse events in fetuses and nursing infants secondary to treatment of the mother with LMY-922 and lymphodepleting chemotherapy. Women of childbearing potential must have a negative serum pregnancy test.
8. Serologic status reflecting active hepatitis B or C infection. Patients that are positive for hepatitis B core antibody, hepatitis B surface antigen (HBsAg), or hepatitis C antibody must have a negative polymerase chain reaction (PCR) prior to enrollment. (PCR positive patients will be excluded.)
9. Patients with history of clinically relevant central nervous system (CNS) pathology such as uncontrolled epilepsy, paresis, aphasia, uncontrolled cerebrovascular disease, severe brain injuries, dementia, and Parkinson's disease.
10. Subjects with uncontrolled intercurrent or psychiatric illness/social situations that would limit compliance with study requirements.
11. Patients receiving a live vaccine within 2 weeks prior to screening.
12. Concurrent use of high dose systemic steroids and/or immunosuppressive therapies.

1. Steroid dose must be weaned to ≤10 mg/day prednisone equivalent prior to CAR-T cell infusion.
2. Immunosuppressive medications must be stopped at least 5 half-lives prior to CAR-T cell infusion.