The Collaborative International Dictionary of English v.0.48:

Batty \Bat"ty\ (b[a^]t"t[y^]), a.
   1. Belonging to, or resembling, a bat. "Batty wings." --Shak.
      [1913 Webster]

   2. Crazy; demented; loony; nuts; as, her constant gabbing is
      driving me batty. [Colloq.]

   Syn: bats.
        [PJC]

WordNet (r) 3.0 (2006):

batty
    adj 1: informal or slang terms for mentally irregular; "it used
           to drive my husband balmy" [syn: balmy, barmy,
           bats, batty, bonkers, buggy, cracked,
           crackers, daft, dotty, fruity, haywire,
           kooky, kookie, loco, loony, loopy, nuts,
           nutty, round the bend, around the bend, wacky,
           whacky]