Can a believer be non-Muslim, and can a Muslim be a non-believer?

Islam is submission to Allah and His commands. It encompasses outward actions, as the Prophet, peace and blessings be upon him, clarified in the Hadith (definition) of Gabriel, whereas faith (Iman) is inner affirmation. Islam and Iman are interconnected and intertwined; faith is internal, and Islam is external, and each can stand for the other when mentioned alone. If they are mentioned together, they differ in meaning, but if they are separated, they converge. The outward actions of a Muslim hold no value unless he believes in his heart, and inner faith is not beneficial if the limbs do not act in accordance with it. A Muslim may be a non-believer if his outward actions do not align with what is in his heart, as is the case with hypocrites, but a believer can only be a Muslim.