i have seen "play" like this between dogs and goats end up in a deadly battle. there was one part of that scene where the dog was on top of the deer and trying to bite it...this can cause a predator to "snap" and go into kill mode...it only takes a second....i found myself wishing the dog would go in the house, and the deer would get outside of the gate...
hope they are both still o.k.

we had a "pet" deer in our neighborhood that gored someone with his antlers...he wandered around for a while with a flea collar on. That
kept people from shooting him when they saw that. he was sent to a
zoo or wild animal park...we called him Bambi...