What evidence is there that New Zealand was part of the British empire?

Answer 1

New Zealand (though there was no such country then) became a colony of the British Empire by default when James Cook et al declared it to be so. (separate declaration for the South Island.)

There us of course, no British Empire these days. Long gone.

There is The Commonwealth, a free association of some of the former colonies, and of some countrieswho were never colonies. Known today for its sports meet, and for educational matters.

Further information:

New Zealand became a member of the British Empire with the signing of the treaty of Waitangi in 1840. The Treaty of Waitangi was an official recognition of the founding of New Zealand by white settlers, and its incorporation into the British Empire. The Treaty was signed on 6 February 1840 by over 500 Māori chiefs of New Zealand and the British Governor William Hobson, representing the British Government. The signing of the Treaty meant that Governor Hobson declared British sovereignty over New Zealand, and the colony of New Zealand was formally proclaimed on 3 May 1840.

Answer 2

There is no longer a British Empire (if there ever was such a body).

New Zealand is a member of The Commonwealth, a group of nations many but not all of which were former colonies. Some members never were British Colonies.