Snevily matching theorem for an elementary abelian group

Let $A$ and $B$ be $k$-element subsets of the additive group of a field of odd characteristic $p$, with $k<p$. There is a bijection $\pi :A\to B$ for which the sums $a+\pi (a)$ are pairwise distinct. The exterior-algebra proof works because $k!\ne 0$ in the field.