Dasatinib in Treating Patients With Unresectable or Metastatic Squamous Cell Skin Cancer or RAI Stage 0-I Chronic Lymphocytic Leukemia

Inclusion Criteria: * Diagnosis of 1 of the following * Histologically or cytologically confirmed squamous cell carcinoma of the skin * Unresectable or metastatic disease * Squamous cell histology represents ≥ 50% of the biopsy specimen * May or may not be related to autologous or allogeneic organ transplantation * Chronic lymphocytic leukemia (CLL) * RAI stage 0-I * Stable disease * Patients with basalosquamous cell disease (basal cell with squamous differentiation) are eligible * Measurable disease, defined as at least 1 unidimensionally measurable lesion ≥ 20 mm by conventional techniques or ≥ 10 mm by spiral CT scan * Must be willing to undergo a pre-treatment tumor biopsy * Brain metastases are allowed provided the following are true: * Received definitive therapy consisting of external beam radiation therapy, gamma knife therapy, or surgical resection resulting in clinically stable disease * Lesions are under control for at least 4 weeks after completion of definitive therapy, as measured by repeat MRI or CT scans * No requirement for dexamethasone * ECOG performance status 0-1 OR Karnofsky 60-100% * Life expectancy \> 6 months * WBC ≥ 3,000/mm\^3 * Absolute neutrophil count ≥ 1,500/mm\^3 * Platelets ≥ 100,000/mm\^3 * Total bilirubin ≤ 1.5 times upper limit of normal(ULN) * AST/ALT ≤ 2.5 times ULN * Potassium 3.5 - 5.1 mmol/L * Calcium \> lower limit of normal * Creatinine ≤ 1.5 times ULN OR creatinine clearance ≥ 60 mL/min * No known HIV 1 or HIV 2 positivity * No known hepatitis C or hepatitis B positivity * Not pregnant or nursing * Negative pregnancy test * Fertile patients must use effective contraception * No history of allergic reactions attributed to compounds of similar chemical or biological composition to dasatinib * No QTc prolongation, defined as a QTc interval of ≥ 480 msecs or other significant ECG abnormality * No condition that impairs the ability to swallow and retain dasatinib tablets (e.g., gastrointestinal tract disease resulting in an inability to take oral medication, requirement for IV alimentation, prior surgical procedure affecting absorption, or active peptic ulcer disease) * No clinically significant cardiovascular disease including the following: * Myocardial infarction within 6 months * Uncontrolled angina within 3 months * Diagnosed or suspected congenital long QT syndrome * Any history of clinically significant ventricular arrhythmias (e.g., ventricular tachycardia, ventricular fibrillation, or Torsades de Pointe) * Any history of second or third degree heart block (may be eligible if the subject currently has a pacemaker) * Heart rate consistently \< 50 beats/minute on pre-entry ECG * Uncontrolled hypertension * Ejection fraction \< 45% by transthoracic echo * No uncontrolled intercurrent illness including, but not limited to, the following: * Ongoing or active infection requiring intravenous antibiotics * History of significant bleeding disorder, including congenital (von Willebrand's disease) or acquired (anti-factor VIII antibodies) disorders * Psychiatric illness or social situations that would limit compliance with study requirements * No prior malignancy except for adequately treated basal cell cancer, carcinoma in situ of the cervix, or other cancer for which the patient has been disease free for 3 years * No gastro-esophageal reflux disease dependent on proton pump inhibitors, H2 blockers, or antacids * Recovered from prior therapy * No more than 1 prior therapy with a monoclonal antibody * No more than 1 prior chemotherapy regimen * No prior tyrosine kinase inhibitor therapy * Prior erlotinib hydrochloride allowed * More than 2 weeks since prior chemotherapy (6 weeks for nitrosoureas or mitomycin C) and recovered * At least 4 weeks since prior radiotherapy * Measurable disease must be outside the radiotherapy port * At least 2 weeks since prior topical therapy * At least 4 weeks since prior surgery requiring general anesthesia and intubation * At least 120 days (4 months) since prior amiodarone * At least 7 days since prior and no concurrent anti-thrombotic and/or platelet agents (e.g., warfarin, heparin, low molecular weight heparin, aspirin \[full dose and 81 mg dose\] and/or ibuprofen) * At least 7 days since prior and no concurrent agents with pro-arrhythmic potential * More than 7 days or 5 half lives, whichever is greater, since prior and no concurrent agents or substances that induce or inhibit CYP3A4 * No concurrent bisphosphonate therapy for the first 8 weeks of dasatinib treatment * No other concurrent investigational agents * No concurrent combination antiretroviral therapy for HIV-positive patients