It was not mentioned in the recent issue of In Our Time which dealt with the unicorn: there is a Unicorn story in the Mahabharata. You can hear that story in the Mahabharata podcast (feed).
Episode 31 - Rshyashrnga, you can hear this story be told by Lawrence Manzo. Manzo also mentions the hypothesis that the story made its way to the West in the form of the Unicorn myth.
The Mahabharata version tells of a Sadhu, a hermit who lives in abstinence, sits by a river after years of not having seen any woman. The river is being visited by some beautiful princess who takes a naked dip and this sight is too much for the Sadhu. He spills his seed, it falls in the water and is drunk by a deer who gives immediate birth to a boy who has an antilope horn growing from his forehead.
The boy grows up with his father the Sadhu who raises him as another hermit. Eventually, the boy, Rshyashrnga, grows up as a formidable yogi who has no knowledge of women. Yet, the moment comes when he is to be seduced and obviously it takes a maiden to conquer the Unicorn.
More Mahabharata Podcast:
The Mahabharata Podcast.