Fur or fleece
I believe the answer is:
coat
I believe this is a double definition.
'fur' is the first definition.
(an animal's coat)
'fleece' is the second definition.
(I've seen this before)
(Other definitions for coat that I've seen before include "Ulster, perhaps" , "blanket" , "Warm outerwear" , "Outer covering" , "Layer of paint" .)