  We study the Yangian $Y_n$ associated to the general linear Lie algebra
$\mathfrak{gl}_n$ over a field of positive characteristic, as well as its
shifted analog $Y_n(\sigma)$. Our main result gives a description of the centre
of $Y_n(\sigma)$: it is a polynomial algebra generated by its Harish-Chandra
centre (which lifts the centre in characteristic zero) together with a large
$p$-centre. Moreover, $Y_n(\sigma)$ is free as a module over its center. In
future work, it will be seen that every reduced enveloping algebra
$U_\chi(\mathfrak{gl}_n)$ is Morita equivalent to a quotient of an appropriate
choice of shifted Yangian, and so our results will have applications in
classical representation theory.
