See note on methodology Since I did the British-ruled non-sovereign territories of Europe a couple of days back, I felt it only appropriate to do the Danish, Finnish and Norwegian equivalents. (NB that Greenland is not in Europe, nor are Ceuta and Melilla, or French Guiana, etc etc.)
The Faroe Islands
There's a very clear winner for the Faroes, a Danish novel about a mild-mannered chap who has a fascination with a particular astronaut. It sounds very entertaining:
Buzz Aldrin, What Happened to You in All the Confusion? / Buzz Aldrin, hvor ble det av deg i alt mylderet?, by Johan Harstad
The Faroes don't do badly in general:
Island of Sheep, by John Buchan
Far Afield, by Susanna Kaysen
Even a couple of novels by Faroese writers:
Barbara, by Jørgen-Frantz Jacobsen
Feðgar á ferð / The Old Man and His Sons, by Heðin Brú
The Åland Islands
Not as rich pickings as the Faroes, but at least the top novel is a recently published best-seller by a native writer:
Is / Ice, by Ulla-Lena Lundberg
Not too far behind is a thriller by a British writer.
The Ice Cage, by Olivier Nilsson-Julien
Bonus islands: Svalbard and Jan Mayen
I'm afraid that I'll have to rule out a well-known children's fantasy because only a small section of the book is set there:
The Golden Compass, by Philip Pullman
Svalbard is reasonably well served, in fact, the top book being a horror story set there just before the second world war:
Dark Matter, by Michelle Paver
See also:
A Woman in the Polar Night, by Christiane Ritter
The Solitude of Thomas Cave, by Georgina Harding