  The Erd\H{o}s-Ginzburg-Ziv constant of an abelian group $G$, denoted
$\mathfrak{s}(G)$, is the smallest $k\in\mathbb{N}$ such that any sequence of
elements of $G$ of length $k$ contains a zero-sum subsequence of length
$\exp(G)$. In this paper, we use the partition rank, which generalizes the
slice rank, to prove that for any odd prime $p$, \[
\mathfrak{s}\left(\mathbb{F}_{p}^{n}\right)\leq(p-1)2^{p}\left(J(p)\cdot
p\right)^{n} \] where $0.8414<J(p)<0.91837$ is the constant appearing in
Ellenberg and Gijswijt's bound on arithmetic progression-free subsets of
$\mathbb{F}_{p}^{n}$. For large $n$, and $p>3$, this is the first exponential
improvement to the trivial bound. We also provide a near optimal result
conditional on the conjecture that $\left(\mathbb{Z}/k\mathbb{Z}\right)^{n}$
satisfies property $D$, showing that in this case \[
\mathfrak{s}\left(\left(\mathbb{Z}/k\mathbb{Z}\right)^{n}\right)\leq(k-1)4^{n}+k.
\]
