Famish likely developed as an alteration of Middle English famen, meaning "to starve." The Middle English word was borrowed from the Anglo-French verb afamer, which etymologists believe came from Vulgar Latin affamare. We say "believe" because, while no written evidence has yet been found for the Vulgar Latin word affamare, it would be the expected source for the Anglo-French verb based on the combination of the Latin prefix ad- ("to" or "toward") and the root noun "fames" ("hunger"). In contemporary English, the verb "famish" is still used on rare occasions, but it is considerably less common than the related adjective "famished," which usually means "hungry" or "starving" but can also mean "needy" or "being in want."
15th century, in the meaning defined at transitive sense 1
Middle English, probably alteration of famen, from Anglo-French afamer, from Vulgar Latin *affamare, from Latin ad- + fames
famish
verb