Old English stempan "to pound in a mortar, stamp," from Proto-Germanic *stampojanan (cf. Old Norse stappa, 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" (cf. Greek stembein "to trample, misuse;" see staff (n.)). The meaning "impress or mark (something) with a die" is first recorded 1560.
Related: Stamped; stamping. To stamp out "extinguish (a fire) by stamping on it" is attested from 1851 in the figurative sense. Stamping ground "one's particular territory" (1821) is from the notion of animals. Italian stampa "stamp, impression," Spanish estampar "to stamp, print," French estamper "to stamp, impress" are Germanic loan-words.