The Straight Dope's blurb on canine transmissible venereal tumor reveals that the oldest mamallian cell-line in the world is an immortal, sexually transmissible zombie tumour.
The really sexy dog STD, though, is something called canine transmissible venereal tumor, a very rare example of what's known as a parasitic cancer. Unlike most other contagious cancers such as cervical cancer in humans, CTVT isn't spread by a virus but (as recently proved) by cancerous cells themselves. Genetic analysis suggests the tumor originated in an individual wolf or domesticated dog, probably in east Asia, between 200 and 2,500 years ago. This long-dead canid's much-mutated cells are still alive and being passed along during coitus (or sometimes through casual contact) centuries later, making it the longest-lived mammalian cell line known. The disease is now found throughout the world, especially where there are large populations of strays. It can be treated with surgery, radiation, and chemo, but most otherwise healthy dogs recover spontaneously after several months.