Phase I Trial of Afatinib (BIBW 2992) and Dasatinib in Non-small Cell Lung Cancer (NSCLC)

Inclusion Criteria: * Pathologically or cytologically documented Stage IIIB/IV non-small cell lung cancer, or unresectable recurrent disease following locoregional treatment. * For Phase 1B Extension Only: * Either or both of the following: A tumor which harbors an activating Epidermal Growth Factor Receptor (EGFR) - mutation; History of objective response, or stable disease for at least 6 months, after treatment with erlotinib, afatinib, or gefitinib. * Either or both of the following: Progression or recurrence of disease after receiving prior continuous gefitinib, afatinib, or erlotinib; A tumor known to harbor a de novo T790M mutation, which is known to confer EGFR TKI resistance. * Participants are allowed to have received systemic chemotherapy or investigational therapy in the intervening period prior to trial enrollment * Capable of giving written informed consent. * Evaluable disease, as follows: For Phase 1A Dose Escalation: Have the presence of any evaluable disease, including bone metastases, effusion, or cystic metastases. For Phase 1B Extension Only: Have progressive and measurable disease as defined by the Response Evaluation Criteria in Solid Tumors (RECIST v 1.1). * Reproductive potential must be either terminated (by surgery, radiation, or menopause) or attenuated by the use of an approved contraceptive method during and for 3 to 6 months following the study. * Participant agrees that IV bisphosphonates will be withheld during the first 8 weeks of dasatinib therapy due to risk of hypocalcemia. * Have recovered from prior drug-related toxicity to Grade ≤ 1 Common Terminology Criteria for Adverse Events (CTCAE) v4, within 21 days of initiation of on-study treatment. * Eastern Cooperative Oncology Group (ECOG) performance status of 0 or 1 at initial enrollment, as assessed by clinician or investigator. Exclusion Criteria: * Have previously completed or withdrawn from this study or any other study investigating dasatinib. Prior treatment with other tyrosine kinases, including afatinib, is acceptable. * Prior recent systemic or investigational therapy within 21 days of initiation of study treatment. An exception is that epidermal growth factor receptor (EGFR) inhibitor may be continued up until 3 days of initiation of study treatment. * Women who are pregnant or breastfeeding. Women of childbearing potential must have a negative pregnancy test (β-HCG test in urine or serum) prior to commencing study treatment. * Patients with documented central nervous system or leptomeningeal metastasis (brain metastasis) at the time of study entry. Patients with prior brain metastasis may be considered if they have completed their treatment for brain metastasis and no longer require corticosteroids. * Patients with disease progression in the central nervous system (CNS) only. * Serious concomitant disorder, including active bacterial, fungal, or viral infection, incompatible with the study (at the discretion of the principal investigator). * Uncorrected severe electrolyte disorder, including severe potassium (\<3.0 mEq/L) or magnesium ( \< 1.0 mEq/L) deficiency. * Any gastrointestinal disorder with diarrhea as a major symptom, such as Crohn's, or pre-existing chronic diarrhea Common Toxicity Criteria (CTC) Grade ≥ 2 of any etiology. Included are malabsorption disorders that in the opinion of the study physician may affect absorption of either afatinib or dasatinib. * Prior major surgery or radiation therapy within 14 days of initiation of treatment. * Electrocardiogram (ECG) abnormalities indicative of arrhythmia (at the discretion of the investigator). * History or presence of clinically relevant cardiovascular abnormalities such as uncontrolled hypertension, congestive heart failure New York Heart Association (NYHA) classification of 3, unstable angina or poorly controlled arrhythmia. Myocardial infarction within 6 months prior to enrollment. * Baseline (\< 1 month before treatment) cardiac left ventricular function with resting ejection fraction of less than 50% measured by multigated blood pool imaging of the heart (MUGA scan) or echocardiogram. * Any history of clinically significant ventricular arrhythmias (such as ventricular tachycardia, ventricular fibrillation, congenital long QT syndrome, or Torsades de pointes). * Prolonged QTc interval on pre-entry electrocardiogram (\> 470 msec for men and \>480 msec for women per American College of Cardiology/American Heart Association \[AHA/ACC\] 2011 scientific statement). * History of significant bleeding disorder unrelated to cancer, including diagnosed congenital bleeding disorders (e.g., von Willebrand's disease). * Patients currently taking drugs that are generally accepted to have a high risk of causing Torsades de Pointes. * Patients with pre-existing interstitial lung disease (ILD), or pericardial / pleural effusion of grade 2 or higher. Trace pericardial or pleural effusion is acceptable. * Patients who require chronic oxygen therapy for chronic obstructive pulmonary disease or pleural effusions (malignant or benign). * Patients requiring comedication with potent P-gp inhibitors (including cyclosporin, azithromycin, erythromycin, ketoconazole, itraconazole, quinidine, phenobarbital salt with quinidine, ritonavir, valspodar, verapamil) or inducers (including rifampicin). * Known active hepatitis B infection, known active hepatitis C infection, or known HIV carrier. * Known or suspected active drug or alcohol abuse. * Known hypersensitivity to afatinib, dasatinib, or the excipients of any of the trial drugs. * Laboratory exclusion criteria: Absolute neutrophil count (ANC) \< 1000 / mm\^3, Platelet count \< 100,000 / mm\^3, Serum creatinine ≥1.5 times the upper normal limit or calculated/measured creatinine Clearance ≤60 mL/min., Total bilirubin ≥1.5 mg/dL (\>26 mol/L, SI unit equivalent), Aspartate amino transferase (AST) or Alanine amino transferase (ALT) ≥ 2.5 times the upper limit of normal (if related to liver metastases ≥ five times the upper limit of normal).