  We construct the geometric Satake equivalence for quasi-split reductive
groups over nonarchimedean local fields, using \'etale Artin-Tate motives with
$\mathbb{Z}[\frac{1}{p}]$-coefficients. We consider local fields of both equal
and mixed characteristic. Along the way, we extend the work of
Gaussent--Littelmann on the connection between LS galleries and MV cycles to
the case of residually split reductive groups. As an application, we generalize
Zhu's integral Satake isomorphism for spherical Hecke algebras to ramified
groups. Moreover, for residually split groups, we define generic spherical
Hecke algebras, and construct generic Satake and Bernstein isomorphisms.
