Animal fur, we're told (4)
I believe the answer is:
hare
'animal' is the definition.
(I have seen 'Swift animal ' mean 'hare' so perhaps 'animal' could also mean 'hare')
'fur we're told' is the wordplay.
'fur' becomes 'hair' (I've seen this in another clue).
'we're told' indicates a 'sounds like' (homophone) clue.
'hair' sounds like 'HARE'.
(Other definitions for hare that I've seen before include "Sect: -- Krishna" , "As mad as this March animal" , "Mammal like a large rabbit" , "animal running in fields" , "Long-eared mammal" .)