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No, hunting is not allowed.
Tramping is allowed, despite the PM saying otherwise.

Yep, she was wrong.
This is from the DOC website:
“Under Alert Level 3 New Zealanders will be able to enjoy a little more of the great outdoors, but DOC’s campsites, huts and visitor centres will...

The PM said today that DOC land was closed for tramping under level 3. But I would not be surprised if that's incorrect.

As Kelton says, I expect it is because private landowners can limit numbers of hunters, and who goes where when. Public land you could end up with 100 people all hunting the same area on the same day.

Yeah, I heard that too, but I wonder if she is correct about that.
If so, I can't even go and put out my game cameras :(

Kind of weird I can't travel 30 minutes or so to hunt on public land, so...