• Age \>= 18 years

• Relapsed or refractory CD19+ B cell malignancies of the one of the following histopathology:

‣ Biopsy proven B-cell non-Hodgkin lymphoma (NHL) of any histopathology (including Richter Transformation of CLL); relapsed or refractory disease defined as:

• Two or more prior lines of therapy, at least one anthracycline containing regimen, unless intolerable. Exception: Patients with Richter transformation of CLL are eligible if they had \>= one prior treatment, including prior BTK inhibition

∙ Demonstration of progressive or stable disease by positron emission tomography/computed tomography (PET/CT) or CT criteria as the best response to the most recent chemotherapy regimen according to the revised Lugano Response Criteria for Malignant Lymphoma.

∙ Measurable disease defined as measurable by CT portion of a PET/CT: To be considered measurable, the must be at least one lesion that has a single diameter of (\>1.5 cm Note: Lesions that have been previously irradiated will be considered measurable only if progression has been documented following completion of radiation therapy

⁃ Biopsy proven SLL or flow cytometry proven CLL; relapsed disease defined as:

• \>= two prior lines of therapy, and/or \>= 6 months of second line prior BTK inhibition (e.g. venetoclax and ibrutinib). Exception: Patients in stable disease (SD) or partial response (PR) with a known ibrutinib resistance mutation (BTK or phospholipase Cgamma2) may be included even if on ibrutinib therapy for less than 6 months.

∙ Demonstration of progressive or stable disease by PET/CT or CT criteria according to the International Workshop on Chronic Lymphocytic Leukemia (iwCLL2018) criteria

∙ Measurable disease by CT portion of a PET/CT where at least one lesion has a single diameter of \>1.5 cm or peripheral blood absolute blood lymphocyte count (ALC) of \> 5000. Note: Lesions that have been previously irradiated will be considered measurable only if progression has been documented following completion of radiation therapy

• Eastern Cooperative Oncology Group (ECOG) performance status (PS) 0 or 1

• Hemoglobin \>= 8.0 g/dL (=\< 14 days prior to registration)

• Absolute neutrophil count (ANC) \>= 500/mm\^3 (=\< 14 days prior to registration)

• Platelet count \>= 30,000/mm\^3 (=\< 14 days prior to registration)

• Total bilirubin =\< 2.0 mg/dL (with the exception of subjects with Gilbert's syndrome. Subjects with Gilbert's syndrome may be included if their total bilirubin is =\< 3.0 x upper limit of normal (ULN) and direct bilirubin =\< 1.5 x ULN) (=\< 14 days prior to registration)

• Alanine aminotransferase (ALT) and aspartate transaminase (AST) =\< 3 x ULN (=\< 14 days prior to registration)

• Prothrombin time (PT) / international normalized ratio (INR) and/or activated partial thromboplastin time (aPTT) =\< 1.5 x ULN OR if patient is receiving anticoagulant therapy and INR or aPTT is within target range of therapy (for patients receiving anticoagulation, there should be no prior history of bleeding, and no recent deep venous thrombosis/pulmonary embolism (DVT/PE) within the last 6 months of enrollment) (=\< 14 days prior to registration)

• Calculated creatinine clearance \>= 45 ml/min using the Cockcroft-Gault formula (=\< 14 days prior to registration)

• Cardiac ejection fraction \>= 50% and no evidence of clinically significant pericardial effusion as determined by an echocardiogram (ECHO) or multigated acquisition scan (MUGA) scan

• Baseline oxygen saturation \>= 92% on room air

• Negative serum pregnancy test done =\< 7 days prior to registration, for persons of childbearing potential only

• Women patients of child bearing potential, including women with tubal ligations, must commit to using use 2 highly effective forms of birth control (defined as the use of an intrauterine device, a barrier method with spermicide, condoms, any form of hormonal contraceptives) for the duration of the study and for 12 months following IC19/1563 therapy

• Provide written informed consent

• Willingness to provide mandatory blood specimens for correlative research

• Willing to return to enrolling institution for follow-up (during the Active Monitoring Phase of the study)