Doraphobia

Doraphobia: An abnormal and persistent fear of fur. Sufferers of this fear avoid fur-bearing animals such as dogs, cats, foxes, beavers and rabbits because fur is repulsive to them. Perhaps some of these phobics associate fur with childhood stories about "the big bad wolf" and other fur-bearing predators. Their fear is not always unfounded, for many furry animals -- including bats, raccoons, dogs and cats -- can be carriers of rabies, an acute viral infection of the central nervous system. An animal can transmit this disease to a human through a bite and, in rare cases, through contact with the animal's blood or tissue or even through inhalation of the rabies virus in a bat cave. The disease is fatal unless an infected person receives rabies shots soon after contracting the illness.

The word "doraphobia" is from Greek roots, namely "dora" (the hide or skin) + "phobos" (fear) = fear of the skin or hide (the fur of an animal).