Sixteen pawns, not one of them able to move, and a result that is already settled.
5 4 
e Now look at what that wall does to the kings. The empty squares on the fourth rank, a4, c4, e4 and g4, are all covered by the black pawns on b5, d5, f5 and h5, so the white king can never set foot on the fourth rank. He is sealed on ranks one to three, and there is not a single black pawn down there for him to eat. Turn the picture upside down and the black king is sealed on ranks seven and eight in exactly the same way: b6, d6, f6 and h6 are covered by white pawns, and a6, c6, e6 and g6 are occupied by his own.
So nothing will ever be captured and no pawn will ever move. The only legal moves left in this game, for the rest of time, are king shuffles. And neither king can even be put in check, because the only units on the board that attack anything are pawns, and a king may not stand on a square a pawn attacks. No check, no mate, not for either player, not in a thousand moves. Article 5.2.2 of the FIDE Laws calls that a dead position and ends the game there, drawn, with no claim and no handshake.
Half a wall is not a wall
This is the position I used to show people, and it is wrong.
7 6 Two pawns locked on one file prove nothing by themselves. A pawn that cannot move can still be taken, and a king that walks round behind it takes it. Play the line through once: the black king cannot both stay in front of the pawn and defend it, White wins it on move four and mates on move eleven. That is the error everybody makes with this rule, and I made it for years: you check the pawns and forget the kings.
A wall counts only when it also shuts the kings out, which is why the position at the top needs every file blocked and every entry square covered.
The test is legal possibility, not good play
The question is never “can the stronger side win this with best play?” It is “does any sequence of legal moves, including terrible ones, end in mate?”
Take the classic wrong rook pawn: White has an h-pawn and a bishop that does not control h8, and Black’s king is already sitting on h8. Every book calls that a draw, and it is: White can never shift him. It is not a dead position. Black could walk his king out to a5, let the pawn queen and be mated a few moves later. Nobody would play that way, but the moves are legal, so the game goes on and the players have to reach the half point some other way.
Same story for a fortress, for opposite-coloured bishops with three pawns each, for almost every position you would call dead drawn in conversation. Those are draws by judgement. A dead position is a draw by arithmetic.
Why you rarely hear it claimed
Because most real ones are already covered by insufficient material, which is the same article doing the same job under a shorter name. Bare kings, king and bishop, king and knight: those turn up every week and get called insufficient material rather than dead positions.
The pawn version is rare enough that an arbiter may want a minute with it. It is still worth knowing for one practical reason: a dead position ends the game at the instant it appears, so nothing that happens afterwards counts. Your flag falling two moves later does not lose you the game, because there was no game left to lose.
The nearest thing you will actually meet is stalemate, which is also instant and also final, and the difference is worth keeping straight: stalemate is one player having no move right now, a dead position is neither player having a mate ever again. Locked walls like the one above grow out of ordinary closed centres, so if you want to learn to read them, pawn endgame puzzles are where they live.