Since I started this series with the component parts of the United Kingdom, I guess I should also note those books set in dependent British territories (and Francis specifically asked about the Isle of Man). A later post will cover the Faroe and Åland Islands.
A clear win for the first book of a Second World War trilogy by a well-known thriller writer:
Night of the Fox, by Jack Higgins
An autobiographical memoir about setting up a zoo is not far behind, but definitely in second place:
Menagerie Manor, by Gerald Durrell
Although Jersey has the larger population, there are a lot more books set on Guernsey. However, it's a decisive win for a wildly successful epistolary novel of the Second World War:
The Guernsey Literary and Potato Peel Pie Society, by Mary Ann Shaffer and Annie Barrows
Strictly speaking the other Channel Islands are politically and administratively part of the Bailiwick of Guernsey, so I don't need to do them (for certain values of "need"). But in for a penny, in for a pound...
Despite extensive research, I'm afraid I am unconvinced that any book set on Alderney qualifies as "well-known". The best I can do is a murder mystery which has some of the action set there, but as far as I can tell most of it is set in Manchester.
Sleep Tight, by Rachel Abbott
Somehow Sark has captured writers much more than Alderney. The winner, I think, is a 1953 short novel by a well-known fantasy writer. Who remembers the TV version starring Derek Jacobi?
Mr Pye, by Mervyn Peake
Honorable mention for a 1989 murder mystery set partly on Sark, partly on Jersey, but as far as I can tell largely in London, much loved by those who have read it:
The Sirens Sang of Murder, by Sarah Caudwell
Not under its own name, but a 1919 novel by the bloke who actually owned Herm is generally understood as being set there. He is better known for his Scottish work.
Fairy Gold, by Compton Mackenzie
Jethou and Brezhou
Nothing to report.
The Isle of Man
The book most frequently tagged "Isle of Man" on LT, certainly much more popular on both LT and GR than any other with a significant Manx component, is a prize-winning 2000 seafaring novel, which however is mainly set in Australia and on the high seas. For the record, it is:
English Passengers, by Matthew Kneale
The top book by ownership actually set on the Isle of Man is a 2012 murder mystery:
Safe House, by Chris Ewan
Gibraltar gets visible pagecount in several well-known books, alas set mainly elsewhere; but they are an eclectic threesome - the conclusion to a best-selling Swedish trilogy; the first in a long series of nineteenth-century naval tales; and the latest novel by a well-known British spy writer (which actually cites my former boss as partial inspiration for the plot). They are:
The Girl Who Kicked the Hornet's Nest, by Stieg Larsson
Master and Commander, by Patrick O'Brian
A Delicate Truth, by John le Carré
The winner for Gibraltar is yet another murder mystery, this time set against the background of a conference of forensic anthropologists:
Uneasy Relations, by Aaron Elkins
Akrotiri and Dhekelia
Again, nothing to report.