What is the national animal of Vatican City?

The Vatican City does not have a national animal.

The Vatican, the world's smallest country, is only 0.82 square miles in size and so cannot sustain any indigenous animals of its own. A national animal is one that is native to that particular country and there are no such species within the Vatican.

What About the White Doves?

We often see The Pope release white doves on special occasions, to symbolize his wish for peace to spread around the world. 

The doves have to be bred specially for their whiteness and are not normally found with this purity of color in the wild. Italian animal rights groups have called upon the Vatican to cease this captive breeding of doves, believing it is cruel and outdated.

This didn't help the Vatican's cause: