Old English stempan "to pound in a mortar," from Proto-Germanic *stamp- (source also of Old Norse stappa, Danish stampe, Middle Dutch stampen, Old High German stampfon, German stampfen "to stamp with the foot, beat, pound," German Stampfe "pestle"), from nasalized form of PIE root *stebh- "to support, place firmly on" (source also of Greek stembein "to trample, misuse;" see staff (n.)). The vowel altered in Middle English, perhaps by influence of Scandinavian forms.
Sense of "strike the foot forcibly downwards" is from mid-14c. The meaning "impress or mark (something) with a die" is first recorded 1550s. Italian stampa "stamp, impression," Spanish estampar "to stamp, print," French étamper (13c., Old French estamper) "to stamp, impress" are Germanic loan-words. Related: Stamped; stamping. To stamp out originally was "extinguish a fire by stamping on it;" attested from 1851 in the figurative sense. Stamping ground "one's particular territory" (1821) is from the notion of animals. A stamped addressed envelope (1873) was one you enclosed in a letter to speed or elicit a reply.