Mo I hit one pretty much straight down one time mid October. It was partially through the shoulder/ one lung. My cousin shot it end of November gun hunting. It's entire body was infected, and puss was coming out of the bullet hole when he shot it. The thing wasn't moving very good, for obvious reasons.
These animals might survive the initial trauma, but infection is a slow, you would think miserable death. That Deers body is under intense distress, if it looks sickly already, it's a dead deer.
I have a story of another bow buck wounded on my farm that was wounded around this time and died around Christmas. I give your deer another 2.5 weeks.