Extract from OED Online

[1935 A. J. POLLOCK Underworld Speaks 53/1 He's bananas, he's sexually perverted; a degenerate.] 1968-70 Current Slang (Univ. S. Dakota) III.-IV. 6 Bananas, adj., excited and upset; 'wild'.College students, both sexes, Kentucky.I'd say it, but everyone would just go bananas. 1970 Times 9 Mar. 43 Liza [Minnelli] moved into the sheltered regimented Barbizon Hotel for Women. Liza says: 'I went bananas!' 1974 Sunday Sun (Brisbane) 3 Feb. 38/3 He just went bananas. My husband tried to take the bottle from him and he wouldn't let go... He jumped onto the television and then onto the china cabinet. 1974 TV Times (Brisbane) 28 Sept. 17/1 'I admit I'm half-bananasnot completely bananas,' he said, 'and the part I play in Dirty Mary, Crazy Larry is a lot like me. Larry is a stock-car racer who goes ape behind the wheel.' 1974 K. MILLET Flying (1975) II. 194 It was driving me bananas, with my sainted mother at the wheel. 1976 Observer 11 Jan. 3/5 She says with her usual verve: 'The Government have gone bananas over a woman tortured in Chile.' 1978 J. KRANTZ Scruples xiv. 412 Jesus, thought Lester, his first movie star and she turns out to be a bit bananas. 1980 Times 1 Oct. 4/1 When the left wing of the Labour Party looks as if it is going to lose, it is described as bananas. 1985 Sunday Times 13 Jan. 5/2 Before we all go bananas about electric cars let me remind you that this is nothing new.