Not quite to the OP, since they were never recognised as independant countries by anyone other than the apartheid-era South African government, but the old “Black Homelands” were virtually all made up of separate parcels of land, acquired by the government as and when they could.
“Islands in the sun” government officials referred to them as - how attractive 
Grim
Yeah, thought of that after I posted it, but I was reasonably sure that someone would correct me. 
Do the Spanish North African enclaves of Ceuta and Melilla count? They are both on the coast and accessible from Spain without crossing another country’s territory so maybe not.
India has the Andaman and Nicobar Islands… separated from the mainland by a few thousand kilometers of Indian Ocean.
Actually, St. Pierre and Miquelon do have Department status; the tourism folk like to pitch St. Pierre as the “last remaining bit of France in North America” (Miquelon is IIRC uninhabited and is a wildlife preserve of sorts). Postal services and law enforcement are handled at least in part by staff that come over from France. When I was there 18 years ago, I got a ride into town from the airport with a couple of gendarmes who were in the midst of a 2-year tour of duty. Seems St. Pierre is considered a hardship post, but they were willing to do it in exchange for the higher pay while on St. Pierre and some additional status when they returned to France.
I also had a lovely experience with getting my bottles of French champagne confiscated when I came back through Canadian customs in Nova Scotia… but that’s another story.
Previous thread: Why does France keep St. Pierre and Miquelon?
does Gaza wrt the West Bank count?
Remember the UAR circa 1958 to 61 ?
I think Pt Roberts would be too small.
Spain and the Canary Islands?
In the spirit of the mandate of the SD, I suggest that the OP should have used the word “contiguous” instead of “continuous” in the title.
West Berlin. Not now, of course, but not so long ago. I wouldn’t think it could be dismissed as a small enclave.
And if Quebec ever decides to pull out of Confederation, Canada will be massively bifurcated.
Ah, we can always replace Quebec with part of the caribbean
Well, the mainland border jogs east to run roughly perpendicular with the border on Tierra del Fuego, which seems to leave Chilean land between the two parts of Argentina (map). I might not be looking at it closely enough, though, and I assume you’re right about Argentina’s territorial waters being contiguous in the area.
Whether the Chilean territory is between them is irrelevant. This is an island, so you’d treat it the same as other islands owned by continental countries. Does Ecuador count since it owns the Galapagos, or Chile since it owns Easter Island? I don’t think the person who posed the question wanted to count them.
As for the territorial waters question, I don’t think you have to leave Argentine waters to reach the Argentine part of Tierra del Fuego from the rest of Argentina. The distance is about 20 statute miles and territorial waters extend 12 nautical miles from each shore.
Actually, no, St-Pierre et Miquelon doesn’t have such a status. It’s an oversea territory, like French Polynesia, New Caledonia, Mayotte, Wallis et Futuna and some small others. A departement is a regular administrative division in France. As such, oversea “departements” are governed in the same way, organized in the same way, regulated by the same laws (with minor exceptions), etc…than mainland “departements”. Put simply, they’re just part of France, though a bit far away from Paris (*).
On the other hand, an oversea territory has a specific status, and these statuses can vary widely. For instance, French Polynesia has an extended internal autonomy, Wallis et Futuna has kings (three of them), in Mayotte, a lot of civil cases are handled by muftis according to the sharia law, etc…(however, AFAIK, St-Pierre doesn’t have a very peculiar status).
However, these territories still have “gendarmes”, french civil servants, most things are regulated by french laws, etc…Which probably explain your mistake. But what a “departement” is is clearly defined and this definition isn’t “it looked quite french when I visited it”.
(*) This allow to ask the following question : “with which country does France has the longest boundary?”. Until now, nobody (I mean french people) found the good answer, even after several tries. The correct one is : Brazil , due to the large size of the Guyanne departement.
Interesting. I was going on my recollection of conversations with the locals when I was there, including a local historian whose book I still have, when I said that St. Pierre et Miquelon was a department. Having also been fortunate enough to visit France on a few occasions, I can truthfully say that St. Pierre doesn’t look terribly French. 
Out of curiosity, I checked the CIA World Factbook entry for St. Pierre et Miquelon. It is in fact defined as a territorial collectivity of France, but the local long form of the country name is “Departement de Saint-Pierre et Miquelon.” One can debate whether the word “departement” should be used if in fact the administration of St. Pierre is handled differently from other departements in France itself, and I certainly defer to your considerably greater knowledge in that regard… but if the locals are going to call it a departement, I’m willing to go along with their usage.
