Another Tic-Tac-Toe problem

Two players A and B play Tic-Tac-Toe on an infinite 3-dimensional grid. Player A starts first and uses X, player B plays second and uses O. Every time player A places an X on the grid, player B can place $n$ number of O's. Player A wins if he eventually succeeds to make 3 in a row - horizontally or vertically, otherwise the game is a draw. For which values of $n$ player A can always win?

Harder version: What if player A has to make $k>3$ in a row? Partial solutions are welcome, 2-dimensional grid and $k=4$ seems doable. (I think the answer is nice, but don't have a complete solution yet)

Remark: No diagonals here.

A wins if $n=2$:
B draws if $n=3$:
The same argument also shows that B can draw on an $m$-dimensional board with $k=3$ if $n\ge m$.