  Let $G$ be a split connected reductive group over a finite field of
characteristic $p > 2$ such that $G_\text{der}$ is absolutely almost simple. We
give a geometric construction of perverse $\mathbb{F}_p$-sheaves on the Iwahori
affine flag variety of $G$ which are central with respect to the convolution
product. We deduce an explicit formula for an isomorphism from the spherical
mod $p$ Hecke algebra to the center of the Iwahori mod $p$ Hecke algebra. We
also give a formula for the central integral Bernstein elements in the Iwahori
mod $p$ Hecke algebra. To accomplish these goals we construct a nearby cycles
functor for perverse $\mathbb{F}_p$-sheaves and we use Frobenius splitting
techniques to prove some properties of this functor. We also prove that certain
equal characteristic analogues of local models of Shimura varieties are
strongly $F$-regular, and hence they are $F$-rational and have pseudo-rational
singularities.
