I shot at a young buck that jumped the string so bad I hit him on the rump, on the other side !
He jumped a few times, stopped and reached back and pulled the arrow out with his mouth, then walked away. It was not buried in him, just more or less under the skin, but it was in him for sure.
I have no idea what pain animals feel, but there is a lot less pain in an arrow cut, or any other cut, than we often attribute. Surface nerves are cut, muscled and internals don't have the same sort of nerves and it just doesn't hurt. Many folks who were shot or stabbed by their own arrow back this up.
ChuckC