  Let $k$ be an algebraically closed field of characteristic $p$. Denote by
$W(k)$ the ring of Witt vectors of $k$. Let $F$ denote a totally ramified
finite extension of $W(k)[1/p]$ and $\mathcal{O}$ the its ring of integers. For
a connected reductive group scheme $G$ over $\mathcal{O}$, we study the
category $P_{L^+G}(Gr_G,\Lambda)$ of $L^+G$-equivariant perverse sheaves in
$\Lambda$-coefficient on the affine Grassmannian $Gr_G$ where
$\Lambda=\mathbb{Z}_{\ell}$ and $\mathbb{F}_{\ell}$ and prove it is equivalent
as a tensor category to the category of finitely generated
$\Lambda$-representations of the Langlands dual group of $G$.
