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Conservation manager Elton Smith said he wanted this to "start a conversation'' around the contentious issue.
"We've had a massive response.''
The picture was taken in March on a camera placed there by a researcher.
The cat would not actually be able to get into the sanctuary, he said.
"Cats are one of the few introduced predators that hasn't been able to get inside. The fence is very cat-proof - but that's not the point.''
The problem was cats killing animals which could get out of the fence, including robins, kaka and various types of lizards, he said.
"We've even had tuatara escape.''
The sanctuary was "absolutely not'' planning to trap cats, he said.
"We couldn't kill them, because clearly we would kill domestic cats. Really, we can't tell the difference.
"We could live trap cats, but still we really couldn't tell the difference. It's too resource demanding and too politically charged.''
Cats were seen "all the time'' roaming around the outskirts of the sanctuary, he said.
"It's the elephant in the room people don't seem to want to talk about.''
Cat owners keeping them inside seemed "almost cruel'' also, he said.
"There must be some middle ground between that and not having a cat walking around a nature reserve at night, which is the highest protection there is for conservation land.''