
No, snow leopards do not live in the Arctic. In fact no cat of any species lives in the Arctic. The climate is just too unsuited and so is the terrain. This majestic large wild cat species lives in Central Asia which is thousands of miles south of the Arctic. That said the habitat of the snow leopard in Central Asia is pretty harsh. It is cold, rocky and mountainous by and large. They prey is blue sheep. The globe below explains things:

Science Daily says that 5,500 animal species live in the Arctic which I find amazing. Hope this helps. There is nothing to add except to say that you can click on this link to see in detail where the snow leopard lives.
P.S. I have just noticed that I have done this post twice! Sorry it happens sometimes when there are so many articles and topics.
Some more on the snow leopard: