 Eligible patients will receive afatinib in treatment cycles of 28 days that will consist of afatinib administered orally by mouth once every four days. Patients will be assigned to the dose level open at the time of their enrollment. Patients will continue dosing of afatinib until disease progression, unacceptable toxicity, withdrawal of consent, or treating physician determines it is in their best interest to stop. Guidelines for modifying study drug doses is provided for the management of adverse treatment effects. 