eyelash Идиома
bat an eye/eyelash
show surprise, fear or interest; show one
bat an eye|bat|bat an eyelash|eye|eyelash
v. phr.,
informal To show surprise, fear, or interest; show your feelings.

Used in negative sentences.
When I told him the price of the car he never batted an eye. Bill told his story without batting an eyelash, although not a word of it was true. Compare: STRAIGHT FACE.