Stuck with animal coat, promises to pay, being very quick (4,3,7)
I believe the answer is:
fast and furious
'very quick' is the definition.
'stuck with animal coat promises to pay' is the wordplay.
'stuck' becomes 'fast' (both can mean secure).
'with' becomes 'and'.
'animal coat' becomes 'fur' (I've seen this before).
'promises to pay' becomes 'IOU's' (more than one IOU).
'fast'+'and'+'fur'+'ious'='FAST AND FURIOUS'
'being' acts as a link.
(Other definitions for fast and furious that I've seen before include "real quick" , "Rapidly and vigorously" .)