Résumé : Let $K$ be a field of characteristic $p>0$ and let $f(t_1,\ldots ,t_d)$ be a power series in $d$ variables with coefficients in $K$ that is algebraic over the field of multivariate rational functions $K(t_1,\ldots ,t_d)$. We prove a generalization of both Derksen's recent analogue of the Skolem--Mahler--Lech theorem in positive characteristic and a classical theorem of Christol, by showing that the set of indices $(n_1,\ldots,n_d)\in \mathbb{N}^d$ for which the coefficient of $t_1^{n_1}\cdots t_d^{n_d}$ in $f(t_1,\ldots ,t_d)$ is zero is a $p$-automatic set. Applying this result to multivariate rational functions leads to interesting effective results concerning some Diophantine equations related to $S$-unit equations and more generally to the Mordell--Lang Theorem over fields of positive characteristic.