Can it be useful for a player block with a hanging piece in a back rank mate situation? Could the following situation possibly exist?

It's White to play, and Black is about to get back rank mated. This is because the King's mobility is restricted to a single file. Then White gives a check.

At this point, Black plays the best move that blocks with a hanging piece, where the blocking piece has no defense. White takes the piece with check, and black can now escape somehow. 

An example of this is that by blocking check with a hanging knight, Black opens up a bishop's path to further block, but this time safely and checkmate is avoided.
[FEN "R5k1/5ppp/8/2r5/1b6/8/5PPP/6K1 b - - 0 1"]

1... Rc8 2. Rxc8+ Bf8


This is an example of what Tim Krabbé calls an 'unguarded guard' - a linepiece checks, and a piece interposes on an unguarded square. That page mentions Topalov - Polgar, Novgorod 1996 (see below); while not a true back rank mate, it comes close. Here are some endgame studies with other types of Unguarded Guards.

[FEN ""]
[Event "Novgorod"]
[Date "1996-07-27"]
[White "Veselin Topalov"]
[Black "Judit Polgar"]
[StartPly "92"]

1.e4 c5 2.Nf3 e6 3.d4 cxd4 4.Nxd4 Nc6 5.Nc3 Qc7 6.Be3 a6 7.Bd3
b5 8.Nxc6 Qxc6 9.O-O Bb7 10.a3 Nf6 11.Re1 Be7 12.Qf3 d6 13.Qh3
h5 14.Bg5 Qc5 15.Be3 Qc6 16.f3 g5 17.a4 b4 18.Na2 g4 19.fxg4
d5 20.c3 bxc3 21.bxc3 Nxg4 22.Bd4 Bc5 23.Be2 Nf6 24.e5 Ne4
25.Nb4 Bxd4+ 26.cxd4 Qb6 27.Nc2 Rc8 28.Rab1 Qa7 29.Bd3 Rg8
30.a5 Bc6 31.Rb6 Bb5 32.Bxe4 dxe4 33.Rxe6+ Kf8 34.Qa3+ Kg7
35.Re7 Rc7 36.Rxc7 Qxc7 37.Ne3 Kh8 38.Rc1 Qd8 39.d5 Qg5 40.Qc3
Kh7 41.Qd4 Bd3 42.d6 h4 43.Rc7 Rb8 44.Rxf7+ Kg8 45.e6 Qxe3+
46.Qxe3 Rb1+ 47.Qc1 Rxc1+ 48.Kf2 Rc6 49.Rd7 Bb5 50.Ke3 Rc2
51.Rc7 Re2+ 52.Kf4 Rf2+ 53.Kxe4 Re2+ 54.Kf5 Rf2+ 55.Ke5 Re2+
56.Kf6 Rf2+ 57.Ke7 Re2 58.d7 Bxd7 59.Kxd7 Rd2+ 60.Ke8 1-0