Upvote:2
You can get a visa. Of course the UK ban might make it tougher since they ask questions on the visa forms but each application is judged on its own merit. For New Zealand, you have to provide solid proof of partnership & you must be of good character (produce a police clearance certificate). They might consider being banned as not of good character. But the UK only bans people for serious issues. Are you sure you were banned.
Upvote:3
Quite a lot of countries will ask you if you ever got removed, anywhere. Telling lies is (a) something which this board does not suggest, on principle, and (b) something whch gets you into even more trouble if you are caught.