Let $F$ be a finite field and $k$ be a positive integer. Let $M_k(F)$ denote the ring of $k\times k$ matrices.
$M_k(F)$ is an $M_k(F)$-module with matrix multiplication, and $F^k$ is an $M_k(F)$-module with matrix-vector multiplication.
Are there any non-trivial $M_k(F)$-modules whose cardinality (the cardinality of the Abelian group) is smaller than $|F|^k$?
The ring $R=M_k(F)$ is simple and artinian. It follows that there exists a unique simple $R$-module $R$ (up to isomorphism) such that every $R$-module $M$ is isomorphic to a direct sum of copies of $S$.
As the module $S=F^k$, with the obvius action of $R$, is simple, it has to be that simple module.
As a consequence of this, the answer to your question is: $|F|^k$.