@Sarg0n I'll not be very surprised if possible, but there are some big problems. First of all, let us distinguish two scenarios: in the final position the king is 1) in check of the pawn that made a move (ordinary checks, orthodox double checks); 2) not in check of the pawn (purely discovered checks, double discovered checks).
In the first scenario the number of squares where the black king can be mated on 6th move is quite restricted. It is definitely on the 7th rank, but moreover, it is from c7 to g7, not a7, b7 or h7, because black king needs to make at least 3 moves to deliver himself there, one of the pieces near the king also needs to move, the pawn whose place the king wants to occupy needs to move and finally the nearby pawn needs to make a move so that it is captured en passant. Total: 6 moves for black, which is unacceptable. For the similar reasons white king must be on c2-g2 if black mates on 5th move by the first scenario.
Next problem: the mating side needs to defend two certain squares by himself: the starting square of the pawn, captured en passant, which is empty in the end and near the king, and the square in front of it, containing the mating pawn. The most natural way to do this is with queen on the 6th rank which needs at least 2 moves to get there (with exception of the square d3 or d6, but then the king is mated on f7 or f2, so if d-pawn is mating, it cannot free the road for the queen beforehand, if it is not mating, it must waste a move, so 2 moves for delivering queen to the right square anyway). If they are not defended by the queen, even more moves are required.
Next problem: either the mating pawn should not be attacked by enemy pieces in the end or it is a double check. Consideration of the probable variants shows that it is very hard for the mated side to block all the required squares for the king in 5 moves, because it is not on the border, clearing all the possibilities of capturing the mating pawn. For the attacking side it is almost no time for depriving the enemy king from some other squares but 2 obligatory above since it has already spent two moves for delivering a queen to the 6th rank and the mating pawn must make three more (leaving only one move for white, probably bishop, but this doesn't help much). If we're mating by a double check, then the second checking piece is definitely a queen, white needs 3 extra moves to defend two obligatory squares with the bishops, so the queen can't move at all, which means the king is mated on d7. But then black hasn't enough time to regroup the pieces as the following example shows: lichess.org/PJqtVrXI
The second scenario (discovered checks) is much more rich and difficult to analyze. There are two squares on the 5th rank that can help to discover a check: of moved pawn and of captured pawn. But the resulting problems are still hard to surmount: if the king is in the open field, it will almost certainly have squares for retreat, if it is in his camp, the defender will almost certainly have opportunities to block the check. The most likely square for checkmate is h6, but I could not come up with something faster than #4.
5.5 moves. Have fun!
I estimated the amount of time a brute force solution would take, and since it seemed feasible, I decided to let a machine solve it for me (with a few improvements over a naive brute force). It did not find any solution for up to 10 half-moves, but it found solutions for 11 half-moves (i.e., same length as your solution). I added one of them to my study: lichess.org/study/NHjJoucl
So the shortest mate is 11 half-moves and there are several of this length even when ignoring move order. The full 11 half-moves search has not been finished yet, so I will maybe post more of them if I find interesting ones among them.
I added a second mate in 11 half-moves to my study (lichess.org/study/NHjJoucl). What I find interesting about that one is that there are two consecutive checks (the second actually is checkmate) by moves of one pawn and both of them are discovered attacks by the same piece, namely the bishop h3 giving check to the king on d7.
Maybe you are also interested in finding the fastest mate by a king move? If you want to look up solutions, you can find a few ones here: lichess.org/study/F8LxRhWJ
François Labelle is arguably the king of such "find a game ending in ___" problems, having spent a long time researching these problems with increasingly-refined computer searches. According to his searches, this particular problem (mate with en passant in 5.5) has 1054 solutions.
Chess problem page of François Labelle: wismuth.com/chess/chess.html
wismuth.com/chess/2kings.html is his article on a unique proof game problem with only 2 kings on the board.
The en passant problem was also published by Hans Klüver in 1967 with 6 solutions listed:
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