Houses across the country are expensive, even in little nowhere towns. Land is sometimes cheap but houses aren't. That's not immigrants doing it, it's the builders and their high wages.
Outside of Auckland, I'd say the problem is more to do with a generation of arrogant local kids growing up thinking they're better than a tradesman and they should get a BA in sociology instead.
To pick on a common scapegoat, Chinese citizens can buy a house in New Zealand, so as their economy improves it drives our housing prices up (same level of supply, greater demand), but I can not buy in theirs.
I think it probably only has a small impact on housing prices (a much bigger problem is the political impossibility of any measure that drops housing prices) but I don't think you can say it has none.
I would personally be in favour of introducing a capital gains tax (but, as a renter, I would say that).