Green’s Dictionary of Slang

(orig. US) to show no emotion, to remain imperturbable.

1894		E.M. Comfort Grizzley's Little Pard 18: Why, she [i.e. a baby] never batted an eye while she was bein’ passed around among a lot of strangers !
1902		Ade Forty Modern Fables 286: He never Batted an Eye when the Peerless Half-Back went down the Field.
1912		Ade Knocking the Neighbors 206–7: Wifey never batted an Eye.
1920		Ade Hand-made Fables 253: Now the Débutante could look straight at any Cardinal Truth and never bat an Eye.
1928		J. Callahan Man’s Grim Justice 23: They could flog me all day long [...] and I’d never ‘bat an eye.’.
1936		‘Banjo’ Paterson Shearer’s Colt 180: Being absolutely impervious to shocks, Mr Manasses never batted an eyelid.
1960		E. De Roo Big Rumble 27: He let one go right in Larry’s face. Larry didn’t bat an eye about it.
2000		Indep. Rev. 12 Jan. 8: The barman didn’t bat an eyelid.