In Forbidden Island, if a player (not pilot or diver) is stranded on a tile with all surrounding tiles sunk, is the game lost or can he be rescued? Can he give away his treasure to a pilot or diver such that other players win but he (personally) dies?
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If a player is stranded on a lone island piece, then they can still be moved using a Helicopter Lift card. Note these can be played by anyone, on anyone's turn, to fly a stranded player to safety (e.g. to Fool's Landing).
If no one has a Helicopter Lift card by the time the island piece needs to be removed from the board, then the player drowns (and everyone loses). On page 7 of the rules under Pawns on Flooded Tiles:
One special case to note is if the tile has to be removed and you have a helicopter card. In this case you can save the sinking player as clarified by Matt Leacock in this Board Game Geek thread.
Note however that you may not use a Sandbags card for this purpose, as specified in the rules:
Remember that all players win or lose together, so if one player drowns but the others survive then it still counts as a loss.
I don't think I agree with tttppp's answer - assuming he is saying that you can't use a Helicopter Lift in response to a tile sinking. The rules for special action cards state:
Yes, there is a specific rule that says you can't use your Sandbag cards reactively to a Flood, but I don't think that implies that Helicopter cards can't be used reactively: quite the reverse if anything.
Other than that I agree with the other answers. You can't win Forbidden Island if any member of your team dies, but there are normally ways out of a tight spot... at least until the waters rise to the point of no return.