The Border-Crosser is drifting through the Caribbean at present. There are 5 integral parts of Europe out here: Martinique, Guadeloupe, French Guiana, St Martin and St Barthelemy. All 5 are part of France: the first 3 are overseas departments, and the last two are overseas collectivities.
St Martin is the most westerly part of the European Union (good quiz tip here). Its status as a collectivity (or COM) is very new - previously it was considered a commune of Guadeloupe. The French St Martin is also unique as it shares the island of St Martin with the Dutch "island area" of Sint Maarten and therefore has a land border with the Netherlands (therein lies another good quiz question about walking from France to the Netherlands without going through Belgium).
The 5 parts of France are all part of the Caraibes co-operation programme (see www.interreg-caraibes.org). This is one of those strange co-operation programmes involving only one Member State, but with a myriad of non-Member States also taking part to some extent. Cohesion funding is only available for the Member State territory (although there is now a 10% flexibility for spending outside the Union), so there are some complicated manoeuvrings to get some degree of access to European Development Funds (EDF - not to be confused with ERDF, which is a Structural Fund for regional development) for the non-Member State islands.
It is a pity that the co-ordination issues are so challenging, as there is a real need for some basic level co-operation on transport (air and maritime), communications, disaster management and environmental issues. It would be nice to think that a fully integrated Caribbean fund could be set up to address these points - but don't hold your breath.