Let $S_n$ be the permutation group and $V = \operatorname{Fun}(X,\mathbb{k})$ functions from $X=\{1,\dotsc,n\}$ to some field $\mathbb{k}$. How can I prove that every irreducible representation of $S_n$ occurs in $V^{\otimes m}$ for integer $m$ large enough?

This is a standard fact when $n!\neq 0$ in $\mathbb{k}$.