A biclique q-coloring is an assignment of q-colors to the vertices of a graph G, so that no biclique (maximal set of vertices that induces a complete bipartite subgraph of G with at least one edge) is monochromatic. Inspired by the coloring game, we introduce the biclique q-coloring game played on a graph G defined as follows. Two players, Alice and Bob, alternately color the vertices of a graph G using q colors. Alice's goal is to color the vertices of G so that no biclique is monochromatic, and Bob tries to prevent this. Both players play optimally and respect the following rule: if a biclique is fully colored, then there exist at least two vertices in the biclique with different colors. In this paper, we prove that the biclique q-coloring game is PSPACE-complete and study the game in powers of paths Pkn.