  The present paper is devoted to studying the super Yangian $Y_{m|n}$
associated to the general linear Lie superalgebra $\mathfrak{gl}_{m|n}$ over a
field of positive characteristic. We extend Drinfeld-type presentations of
$Y_{m|n}$ and the special super Yangian $SY_{m|n}$ to positive characteristic.
Moreover, the center $Z(Y_{m|n})$ of $Y_{m|n}$ is described: it is generated by
its Harish-Chandra center together with a large $p$-center. We also study the
$p$-center of $SY_{m|n}$ and provide another description of the $p$-center of
$Y_{m|n}$ in terms of the RTT generators.
