Trapped by responsibility, backing flier (4)
I believe the answer is:
ibis
'flier' is the definition.
(ibis is a kind of birdanimal)
'trapped by responsibility backing' is the wordplay.
'trapped by' says the answer is hidden in the clue.
'backing' is a reversal indicator.
'sibi' is hidden in the letters of 'responsibility'.
'sibi' backwards is 'IBIS'.
(Other definitions for ibis that I've seen before include "Wader once sacred in Egypt" , "Wading bird sacred to the Egyptians" , "leggy bird" , "Hotel group" , "Wading bird of warm regions" .)