Green’s Dictionary of Slang

not bat an eye(lid) v.

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

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 !
[US]Ade Forty Modern Fables 286: He never Batted an Eye when the Peerless Half-Back went down the Field.
[US]Ade Knocking the Neighbors 206–7: Wifey never batted an Eye.
[US]Ade Hand-made Fables 253: Now the Débutante could look straight at any Cardinal Truth and never bat an Eye.
[US]J. Callahan Man’s Grim Justice 23: They could flog me all day long [...] and I’d never ‘bat an eye.’.
[Aus]‘Banjo’ Paterson Shearer’s Colt 180: Being absolutely impervious to shocks, Mr Manasses never batted an eyelid.
[US]E. De Roo Big Rumble 27: He let one go right in Larry’s face. Larry didn’t bat an eye about it.
[UK]Indep. Rev. 12 Jan. 8: The barman didn’t bat an eyelid.