Do individual mammal hairs absorb water?
Researching this answer took me to some strange literature, but it appears to suggest a little bit of both (at least for human hair, which is porous, implying that some water can suffuse through the cuticle into the cortex). For example this paper looks at the impact of this effect on drug testing. 

There's also a few papers in cosmetics science journals describing various elements and effects of increased hair porosity, but I was unable to find one primary citation which characterised the trait.