Crossword Genius

Run off, chase after – has dropped the coat (6)

Ross

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" .)

I've seen this clue in The Telegraph.
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