I'd say that depends completely on that existing country's willingness to give up part of its territory. As it is, pretty much all of them don't like that idea at all.
I wouldn't bet on it, but I think some of around 200 countries might want to give up a small part of its territory for an appropriate sum of money.
The only countries who might be willing to do that (tinpot little dictatorships, presumably) are also the ones who are likely to claim it back after a couple of years. ("What? We never sold you this island!")
Better not bet on it :) There's a big difference between selling the legal ownership to land and giving up all sovereign rights to it.
I heard Greece could use some money
Germany actually suggested that Greece sell some island to pay their debt - more of snub than a real comment though I think
Even though, they'll still belong to Greece, it's the same as buying any other property. Just a bit more expensive.