Fur left behind by old pet (3)
I believe the answer is:
cat
'pet' is the definition.
(cats are often kept as pets)
'fur left behind by old' is the wordplay.
'fur' becomes 'coat' (an animal's coat).
'left behind by' indicates named letters should be taken away.
'old' becomes 'o' (abbreviation).
'coat' with 'o' taken out is 'CAT'.
(Other definitions for cat that I've seen before include "Common pet" , "Domestic animal" , "Sailor's whip with nine knotted strands" , "An old whip with nine tails" , "Feline - whip" .)