Let $S$ be an irreducible projective surface defined over $\mathbb{Z}_p$, denote its special fibre by $S'$, and let $\gamma: S\longrightarrow S$ be an automorphism which is compatible with the reduction map, having the property that the restriction of $\gamma$ to the special fibre is the identity.
Furthermore, assume the pullback of $\gamma$ to the geometric special fibre is still the identity, and moreover, for any geometric special point $\overline{x}\in S'(\overline{\mathbb{F}_p})$, the residue disk $\mathcal{D}_{\overline{x}}\subseteq S(\overline{\mathbb{Q}_p})$ has a fixed point.
Is it true that $\gamma$ must be the identity on $S$?