1930, in reference to naval build-ups, from arms (see arm (n.2)) + race (n.1). First used in British English.
Entries linking to arms race
arm n.2
[weapon], c. 1300, armes (plural) "weapons of a warrior," from Old French armes (plural), "arms, weapons; war, warfare" (11c.), from Latin arma "weapons" (including armor), literally "tools, implements (of war)," from PIE *ar(ə)mo-, suffixed form of root *ar- "to fit together." The notion seems to be "that which is fitted together." Compare arm (n.1).
The meaning "branch of military service" is from 1798, hence "branch of any organization" (by 1952). The meaning "heraldic insignia" (in coat of arms, etc.) is early 14c., from a use in Old French; originally they were borne on shields of fully armed knights or barons. To be up in arms figuratively is from 1704; to bear arms "do military service" is by 1640s.
race n.1
[act of running] late Old English, also rase, "a narrative, an account;" c. 1300, "an act of swift running, a hurried attack," also "a course of life or conduct, a swift current;" from Old Norse rās "a running, a rush (of water)," cognate with Old English ræs "a running, a rush, a leap, jump; a storming, an attack;" or else a survival of the Old English word with spelling and pronunciation influenced by the Old Norse noun and the verb. The Norse and Old English words are from Proto-Germanic *res- (source also of Middle Dutch rasen "to rave, rage," German rasen, Old English raesettan "to rage" (of fire)), from a variant form of PIE *ers- (1) "be in motion" (see err).
Originally a northern word, it became general in English c. 1550. Formerly used more broadly than now, of any course which has to be run, passed over, or gone through, such as the course of time or events or a life (c. 1300) or the track of a heavenly body across the sky (1580s). To rue (one's) race (15c.) was to repent the course one has taken.
Meaning "contest of speed involving two or more competitors; competitive trial in running, riding, etc." is from 1510s. For the sense of "artificial stream leading water to a mill, etc.," see race (n.3). Meaning "electoral contest for public office" is by 1827.