Whilst I am sure ‘play’ does exist in the animal kingdom in the sense of: things done for no other purpose than individual entertainment, a lot of what you’re seeing when you watch cats, dogs etc ‘play’ is practice for hunting or mating, plus jostling for social status within the group.
All of the “bigger animals” you list are mammals, and if you were to expand out from mammals, your next several would probably be birds. Mammals and birds are all fairly K-selected, which means that the parents have relatively few offspring, but invest significantly into their success. A fair part of that investment is teaching and learning how to perform basic tasks, and it also involves at least some degree of social structure. Play is involved in both of those.
Once you get into R-selected animals (many offspring left to fend for themselves in the hope that a few survive), like most fish or reptiles, I would expect a much different pattern. R-selected animals have to know how to do whatever it is they do instinctively, without learning, and usually aren’t social, because they’re competing directly against their siblings. So there’s no place for play there.