No; there are classes and ways to get companion creatures; but strictly rules wise It’s a bit iffy if you can just “Tame” a beaten creature.
This is up to debate (and thus up to your DM) but if you are Trained in Nature; somewhere it lists the things you can do with it, and I believe it says you can try to clam animals into trusting you. From there it’s up to the DM what else can be done.
Note: Either Shaman or Druids have the ability to speak to animals, as well there is a Monstrous race of animal shape shifters (had an Asian name to it) which can also speak to animals. You could make a good argument that with a high Nature check, as well with the ability to converse with an animal’s might allow to “tame” something.
If I was a DM though I would have the “tame” creature just an NPC that follow the party under DM’s control.