Or one of them, anyway; the Duke, Federico II Gonzaga, loved dogs and owned over a hundred of them when Titian painted this portrait around 1525. This page makes the claim that this silky-looking fellow is a Havanese. Since the Havanese is said to be Cuba's only native breed, and Spain controlled Cuba at this time and had for a generation, I should think this Italian duke could have gotten a pup for a very special gift. Why not - giraffes and elephants have been gifts among the mighty as well, though I bet they don't cuddle as winsomely as Federico's friend.
This work is oil on canvas, and I am sure it's much more splendid in person than in reproduction (thanks anyway Wikipedia). You may see it if you are at the Prado in Madrid.