Chase catch and punish? No! She will just think “sod going back to the human, I just get told off for that” she won’t distinguish between the two events (coming back or chasing deer) she will associate the telling off with the last event. In this case, you catching her!
Two things to keep in mind from my experience.
One, the deer at that moment in time is the most entertaining/intoxicating thing going. You simply need to offer a better alternative. You just haven’t found d it yet. For our terrier Eddie (who lives to chase wildlife) it’s his ball that wins every time.
Second is showing your displeasure at your dog’s interest in deer before she starts the chase. Try her on the lead in an encounter with a deer. Then you have the chance to pull her in and say no at the moment she notices the deer. You could also try to treat her/positively reinforce the good behaviour when you see she is actively ignoring the deer (again, on the lead)
You are doing the right thing in going to the gun dog trainer though. He will know what he’s on about (unlike us wannabes!)
Good luck with her!