Yes they would be a kiwi. Our daughter was born here last year and we have a NZ passport for her. We have also returned to the UK on that passport with no problems (she was not allowed to work by the stamp in her passport but she was 13 months so not a big issue
). We haven't got a UK one yet as it is very expensive here. We will probably wait until/if we return to the UK. One difference our daughter will have is she is deemed British by descent and as such any children she may have in the future would not automatically be entitled to a British passport