  We show that the nearby cycles functor for the $p$-adic Hecke stack at
parahoric level is perverse t-exact, by developing a theory of Wakimoto
filtrations at Iwahori level, and that it lifts to the $\mathbb{E}_1$-center.
We apply these tools to construct the Arkhipov-Bezrukavnikov functor for
$p$-adic affine flag varieties at Iwahori level, and prove that it is an
equivalence for type $A$ groups.
