This would be a bit ironic. It would be hard, but I personally don't see a reason why it wouldn't be possible. As @Richard said, a dementor is the embodiment of unhappiness, but there are some people who feel happy when they are depressed. There are stranger ways people can feel/behave.
Another issue with a Patronus being a dementor is why the person creating this Patronus would need to create one, if he/she had an affinity with dementors. In this case, I would imagine he/she wouldn't have many happy memories to draw on, though, again, there are stranger things in this world than a person being happy and depressed at the same time. Also, there are other creatures that get repelled by Patronuses, so even if the person didn't need to create one to battle dementors, he/she may need one for these creatures. Also, there is the matter of creating one for fun/training, as Dumbledore's Army did in Order of the Phoenix.
Regarding a dementor being an animal, I'm not sure, but as they are (slightly) intelligent, I don't think they are "merely" animals, in the same way humans aren't "merely" animals. That being said, I don't believe J.K. Rowling ever said a Patronus has to be an animal. I don't see why, for example, a Patronus couldn't be a human, or a giant, etc. For narcissists, like Gilderoy Lockhart, I imagine they could have a Patronus in the form of themselves.