"unawares," by 1936, American English, from off (adv.) + base (n.); a figurative extension from baseball sense of a runner being "not in the right position" (1882) and vulnerable to being picked off.
off-base housing
of-
ofay
off
offal
off-and-on
off-base
offbeat
off-Broadway
off-camera
off-chance
off-color