Green’s Dictionary of Slang

angora n.

1905		Sun. Times (Perth) 3 Sept. 4/7: D’ye think a man's a bleedin’ Angora, y’ flamin' lump of carnal iniquity.
1914		Ade ‘The New Fable of the Private Agitator’ in Ade’s Fables 14: Every time he was given a Crimp in the Rue de la Paix he caught even by leading a new Angora up the Chute and into the Shambles.
1947		C. Mann Light in the Valley 166: [a gun shot] ‘Where the deuce did that come from?’ ‘Aw some angora on th’ boat havin’ a potshot at the cormorants.’.

In phrases

get someone’s angora (v.) (also get someone’s fur up) [a pun on get someone’s goat under goat n.¹ ]

(US) to annoy, to irritate.

1912		Eve. World (NY) 20 Feb. 1: He wasn’t afraid to beard lion in den, but Miss Rhino gets his angora. ‘Never again!’ is Zoo man’s comment when rescued from irate beast.
1916		Bridgeport Eve. Farmer (CT) 2 June 15/7: They [i.e. boxers] all get grouchy just bvefore a fight [...] Freddie will work like a horse for weeks, but the last two or three nights get his angora.
1919		F. Hurst ‘White Goods’ in Humoresque 147: Gad! that old hag gets my fur up.
1922		M.E. Smith Adventures of a Boomer Op. 45: That morning in Great Falls got my Angora.
1926		Wood & Goddard Dict. Amer. Sl.