Player A has a winning strategy. I assume $k$ is fixed, and we can replace $5$ with any number. So let $G_{n,k}$ be the game where player A plays one integer each turn and player B plays $k$ integers each turn, with all of them distinct, and player A wins when there is a length $n$ arithmetic sequence amongst his numbers.
This game is open for player A, since every win happens at a finite stage of play if at all. So it is determined by the Gale Stewart theorem.
But in fact, the winning strategy for player A is simply to play the smallest available integer. It follows that in infinite play, every integer will be played, and the set played by A will have proportion at least $1/(k+1)$ in any initial segment. It follows by Szemerédi's theorem that the set contains arbitrarily long arithmetic progressions. So player A will have won at some stage.