Depends whether we're talking about the furry kind or the human kind! Human, there were plenty, but the furry, i'm not aware of any either. Possibly surprising though as, in the early part of the century, it was very fashionable to have a monkey for a pet. My grandmother used to have a Chipmonk - horrible thing used to sit on her shoulder. Not that I was around then, but we have family photos of the thing.
There was another small dog called Freu Freu. Newlyweds Helen and Dickinson Bishop bought it whilst on their honeymoon to Egypt, Italy, France and Algiers (assume it was bought in France with a name like that). However, she was forced to leave it in here state room when they were called to the lifeboats. Apparently Freu Freu was the only dog allowed to stay in a room due to the fact that the steward though she was so cute and pretty.
I hope JJ Astor did let the animals out, or someone else, because they must have been absolutely terrified and not understood why. At least humans have that advantage.
Personally my dog's part of the family, not just our pet.
Regards,
Andrea