A class of games, of which Tic-tac-toe(2,3) is the most popular

Tic-tac-toe(n,k) is played in a discrete n-space. On his turn, a player claims an unclaimed vector of dimension n with each coordinate in the set {1,...,k}. If at the end of his turn the player owns a set of k colinear vectors, that player is declared the winner and the game terminates.

I have hypothesized that if k is less than n+1, then the first player will always win if he is competent.