Here is the advice from "The Complete Dressmaker", published in 1907:
CLEANING FUR
The dark furs, such as seal, mink, sable, etc., respond to mahogany or fine cedar sawdust as a cleanser. These may be purchased from any furrier. Place the fur on a table with the hairy side up and rub sawdust in by the handful. Use plenty of sawdust and rub vigorously. Shake the fur over the table to save the sawdust that fall, as it can be used again. Turn the fur with the hair side down on large pillows according to the size of the garment; beat it well with a switch. Shake the pillows occasionally and continue beating until all the sawdust is removed. White furs are cleaned in a similar fashion with corn meal.
Here is the furriest photo I could find at the moment. She is one of a pair Rat Masks, made for a BayCon many, many years ago. I still like the masks, the paws, tail, and feet, but the rest of the costumes were rather pathetic. I hadn't started doing any historical research at that point. When I look back, I cringe. Someday I need to make some new clothes for my Rodents.