Could you please refer some examples of game where, on the last move, a pawn checkmates the king (preferably from grandmaster's games)?
One amazing game I know that ends in a pawn mate in one called The Polish Immortalm in which Black sacrifices all four minor pieces to win the game! The pawn does a double-step to give the mate.
[Title "Glucksberg-Miguel Najdorf, Warsaw Poland, 1929, The Polish Immortal"] [FEN ""] 1.d4 f5 2.c4 Nf6 3.Nc3 e6 4.Nf3 d5 5.e3 c6 6.Bd3 Bd6 7.O-O O-O 8.Ne2 Nbd7 9.Ng5 Bxh2+ 10.Kh1 Ng4 11.f4 Qe8 12.g3 Qh5 13.Kg2 Bg1 14.Nxg1 Qh2+ 15.Kf3 e5 16.dxe5 Ndxe5+ 17.fxe5 Nxe5+ 18.Kf4 Ng6+ 19.Kf3 f4 20.exf4 Bg4+ 21.Kxg4 Ne5+ 22.fxe5 h5#
Although it was a blitz game, this game ends in a nice discovered checkmate by a pawn.
[Title "Ely Sollano-Fedrick Rhine, Chicago USA, 1977"] [FEN ""] 1. d4 Nf6 2. Nc3 d5 3. Bg5 c5 4. Bxf6 exf6 5. dxc5 d4 6. Ne4 Bf5 7. Ng3 Bxc5 8. Nxf5 Qa5+ 9. c3 dxc3 10. b4 Bxb4 11. Qc2 Qxf5 12. Qxf5 c2#
Another pawn move that results in checkmate is promotion. While there are many such games that end with it, this game has a unique perk in that the mating move was done by the pawn that made the first move.
[Title "Bojan Kurajica-Michael Franklin, Hastings England, 1972"] [FEN ""] 1.e4 c5 2.Nf3 Nc6 3.d4 cxd4 4.Nxd4 e6 5.Nc3 Qc7 6.g3 a6 7.Bg2 Nf6 8.O-O Nxd4 9.Qxd4 Bc5 10.Bf4 d6 11.Qd2 h6 12.Rad1 e5 13.Be3 Bg4 14.Bxc5 dxc5 15.f3 Be6 16.f4 Rd8 17.Nd5 Bxd5 18.exd5 e4 19.c4 O-O 20.Rfe1 Rfe8 21.Qc2 Re7 22.Bxe4 Ne8 23.Bd3 Nd6 24.Re5 Rde8 25.Rde1 f6 26.R5e2 Kf8 27.Re6 Rxe6 28.dxe6 Ke7 29.Bf1 Kd8 30.Bg2 Qa5 31.Qf2 Ke7 32.b3 Nf5 33.Bxb7 Nd4 34.Bd5 Rd8 35.Rd1 Qb6 36.Qe3 a5 37.Qe4 a4 38.Qg6 axb3 39.Qf7+ Kd6 40.e7 Rd7 41.e8=N#
[Title "Gunner Gundersen-A. H. Faul, Pietzcker Christmas Tournament, Melbourne Australia, 1928"] [FEN ""] 1. e4 e6 2. d4 d5 3. e5 c5 4. c3 cxd4 5. cxd4 Bb4+ 6. Nc3 Nc6 7. Nf3 Nge7 8. Bd3 O-O 9. Bxh7+ Kxh7 10. Ng5+ Kg6 11. h4 Nxd4 12. Qg4 f5 13. h5+ Kh6 14. Nxe6+ g5 15. hxg6#
Lastly, although it is not from a game, here is a very pretty chess problem in which White sacrifices all their pieces and a pawn to give mate with the last unit-a pawn!
[Title "Johann Christoffel Van Gool, Le Journal de Genève 1979, Mate In 12 Moves"] [FEN "2n1QN2/2Np1pk1/1b1p1p2/1Kp1B1p1/1R3p2/2P1r3/1r1nPRb1/1B5q w - - 0 1"] 1. Nfe6+ dxe6 2. Nxe6+ fxe6 3. Bxf6+ Kxf6 4. Rxf4+ gxf4 5. Qf8+ Ke5 6. Qxf4+ Kd5 7. Qg5+ Re5 8. e4+ Bxe4 9. Qxd2+ Rxd2 10. Ba2+ Rxa2 11. Rd4+ cxd4 12. c4#
A beautiful, more recent, game between Svilder and Carlsen:
[Title "Svidler - Carlsen, GRENKE Chess Classic, 2019"] [FEN ""] 1.e4 c5 2.Nf3 Nc6 3.Nc3 e5 4.Bc4 Be7 5.d3 d6 6.Nd2 Nf6 7.Nf1 Nd7 8.Nd5 Nb6 9.Nxb6 axb6 10.c3 O-O 11.Ne3 Bg5 12.O-O Kh8 13.a3 f5 14.Nxf5 Bxc1 15.Rxc1 Bxf5 16.exf5 d5 17.Ba2 Rxf5 18.Qg4 Rf6 19.f4 exf4 20.Qg5 Qf8 21.Qxd5 Rd8 22.Qf3 Ne5 23.Qe4 Ng4 24.Rce1 Ne3 25.Rf2 Re8 26.Qxb7 g5 27.Rfe2 g4 28.Rf2 Qh6 29.Qc7 Ref8 30.h3 gxh3 31.g3 fxg3 32.Rxf6 h2+ 33.Kh1 g2#