Run off, chase after has dropped the coat (6)
I believe the answer is:
fleece
'the coat' is the definition.
(fleece is a kind of coat)
'run off chase after has dropped' is the wordplay.
'run off' becomes 'flee' (I've seen this before).
'chase after' becomes 'chase' ('chase' can be a synonym of 'chase after').
'dropped' indicates named letters should be taken away.
'chase' with 'has' taken out is 'ce'.
'flee'+'ce'='FLEECE'
(Other definitions for fleece that I've seen before include "Sheepskin - strip - swindle" , "warm garment" , "Rip off" , "Overcharge" , "Rob by trickery, say woollen coat" .)