/u/meinthemirror answers a closely related question: If beaver pelts were so highly sought after from the "new world," why wasn't the North American beaver domesticated along the lines of the fox and mink fur trade? Was there really that big on an abundance in the wild, or was there some other reason?

The summary is that beavers, at least, were really difficult to domesticate. They have very specialized habitat and dietary requirements, and it probably wouldn't have been cost-effective to domesticate them.

What would it cost to transport, breed, and raise a wild animal population in a farm or ranch like setting to point of commercial viability back then? Not very many animals in the history of mankind have received this treatment, and it usually takes quite a bit of time.