late 14c., infant, infaunt, "a child," also especially "child during earliest period of life, a newborn" (sometimes meaning a fetus), from Latin infantem (nominative infans) "young child, babe in arms," noun use of adjective meaning "not able to speak," from in- "not, opposite of" (see in- (1)) + fans, present participle of fari "to speak," from PIE root *bha- (2) "to speak, tell, say." As an adjective in English, 1580s, from the noun.
The Romans extended the sense of Latin infans to include older children, hence French enfant "child," Italian fanciullo, fanciulla. In English the word formerly also had the wider sense of "child" (commonly reckoned as up to age 7). The common Germanic words for "child" (represented in English by bairn and child) also are sense extensions of words that originally must have meant "newborn."