In the Prisoner of Azkaban, Sirius in his Animagus form, after driving away Lupin in his werewolf form, goes after Pettigrew who escaped in his Animagus form in the meantime. But Harry finds him shortly thereafter turned back into human form and crouching with his hands over his head, all apparently because of the Dementors swarming around.
Why did Sirius not fight the Dementors? Did he not have a Patronus? Sirius is shown in the series to be, along with James Potter, an extremely talented wizard. If James and Lupin had their own corporeal Patronuses (or Patroni?) why did Sirius not master the charm?
Just before his transformation, Sirius had been using Snape's wand. It has been answered previously in this site that Animagi could retain their (clothes and hence) wands in animal form. It also seems to be a consensus that Pettigrew retained Voldemort's wand in Animagus form. Thus, it was definitely possible for Sirius to have retained Snape's wand when he transformed.
Moreover, he was forced out of his animal form, a not so easy feat, which in case of Pettigrew a little earlier, had required the combined magical might of Sirius and Lupin. It cannot be that Dementors always have this prowess, since Sirius is the man who escaped Azkaban using his Animagus form.
Of course, there were a lot of Dementors there, but there were Harry and Hermione in the scene too, and even the latter who was untrained in the matter didn't pass away as easily or as soon. And Sirius is the man who is known to have defied the Dementors right inside Azkaban.