The Secret of Spiggy Holes

The children, reunited with their parents again (and Jack now adopted as part of the family), go on holiday at the rugged Cornish coast with a family friend, Miss Dimity ("Dimmy", who would appear in several further books in the series), and get drawn into a kidnapping case as they discover a boy being held prisoner in a castle tower nearby.
The boy turns out to be the Prince of Baronia, Paul, whom they rescue in due course, and protect as the kidnappers come after him in pursuit. Baronia is a small, remote kingdom of unspecified location (possibly eastern European), and Paul is heir to the throne. The Arnold children become firm friends with Paul during this adventure, and he appears in each of the remaining books in the series.

In the course of the pursuit by the kidnappers, eager to get hold of him again, the five children visit the Secret Island from the previous book again, and hide there for some time. Paul's father, the King of Baronia, is very grateful.