A breed of goat that has a genetic defect called myotonia congenita in its muscle proteins, such that when it's startled its muscles lock up and it falls over. It was used to protect herds of sheep: as soon as the herd was attacked by a wild dog, the sheep scattered but the goat lay there begging to be eaten. The breed rapidly dwindled towards extinction.

In recent years their fortunes have revived and they are kept as novelty pets. A movie clip of a goat being chased and fainting is downloadable at the website of the American Tennessee Fainting Goat Association (founded 1987):

http://www.webworksltd.com/webpub/goats/faintinggoat.html

They are small, but not pygmy, and because they learn that leaning against a wall is safer than vigorous exercise, they tend to be more chubby and meaty than other breeds of comparable size. Because they are so calm they make good pets. They have unusually bulgy eyes and very long protuberant ears. Their coat comes in a variety of colours, though many breeders favour black and white.

The original stock is said to derive from Marshall County, Tennessee in the 1880s, where a mysterious stranger called John Tinsley arrived with four of them. He might have come originally from Nova Scotia but it is not known where he got his goats. He married, sold his goats to a local, and disappeared, leaving behind his wife but taking with him his sacred cow by the name of Zebu.

Other names for the breed include nervous, stiff, myotonic, and Texas wooden leg. The variety transplanted to Texas had been bred more for meat quality, and when interest in these endangered varieties sparked in the 1980s, the Tennessee variety was contrastingly bred smaller and stiffer as a novelty pet.

More at:
http://www.albc-usa.org/tenngoat.htm (conservancy)
http://www.angelfire.com/tx/bgoat/