I wouldn't think cats had patience for going into space (though they have had their place in aviation history). However, in October 1963 the French sent a cat up in a rocket from their base in the Algerian Sahara. Depending on where you read the history, the catonaut was either a black-and-white street fellow named Felix, or a girl cat named Felicette. It seems Felix may have taken a powder or just didn't have the right stuff at the last minute.
Whichever explorer it was, cat and rocket went about 130 miles into space, and then after 15 minutes the capsule fell away and the pod was retrieved, kitty safe and probably beside itself inside. Unfortunately, a second attempt a week later did not go so well, and that cat did not survive.
This whole thing of sending animals into space was always distasteful to me. The purpose was to test the effect of such adventure on vertebrates without risking human beings, and I suppose that's appropriate, but I just never liked the idea. (Let's face it, don't most of us have a short list of people they could have experimented with instead?)
Here's a good link about this little piece of space history, with source links to look over. Felix made it to a stamp. Not much consolation to Felix I bet.