What is the northernmost city in Norway?

Depending on what you mean by city, it could be Longyearbyen, the only incorporated town in Svalbard, at 78:22 north. The population is under 2000, but it used to be called Longyear City.
On tighter definitions of city & Norway, Honningsvåg (Designated a town on 1996, but <5000 people) or Hammerfest (under the current Norwegian rule of >5000 people) .