Green’s Dictionary of Slang

absolutely crazy, highly eccentric; usu. as barking mad.

1900		Bulletin (Sydney) 29 Sept. 31/1: Hochblatter turned the corner and ran into a couple of constables, who promptly ran him in as a lunatic who had got loose in his night-gown, for what with anxiety for his furniture, excitement, unwonted exertion and a foreign tongue, the flying apparition was incoherent, and the watch-house-keeper entered him on the books as ‘barking-mad.’.
1989		Viz June/July n.p.: Barking mad! Has the Queen lost her marbles?
1991		J. Osborne Déjàvu Act I: Cliff: J.P.’s fairly potty at present. Aren’t you? J.P.: Barking.
1994		J. O’Connor Secret World of the Irish Male (1995) 39: Prince Charles is barking anyway. He is quite simply several Chevrolets short of a funeral.
1994		Liverpool Echo 18 Mar. 48/3: [advert] These prices are so totally stark staring raving barking mad. It is quite obvious the manager [...] is off his trolley.
2000		Observer Rev. 20 Feb. 16: An intelligent and not-obviously-barking New Yorker.