Let $\mathfrak{X}$ be a $p$-adic flat formal scheme over $\mathbf{Z}_p$, whose special fiber has an ample line bundle. Then $\mathfrak{X}$ is algebraizable, that is there exists an algebraic $\mathbf{Z}_p$-scheme $X$ such that its $p$-adic formal completion is canonically isomorphic to $\mathfrak{X}$ as a $p$-adic formal scheme, and is unique up to isomorphism.
Grothendieck's Formal Existence Theorem implies the formal completion functor induces an equivalence of categories
$$\text{Coh}(X)\simeq\text{Coh}(\mathfrak{X}).$$
Does it induce an equivalence of étale sites $X_{\rm ét}\simeq\mathfrak{X}_{\rm ét}$, resp. finite étale sites $X_{\rm fét}\simeq\mathfrak{X}_{\rm fét}$? Do their $\ell$-adic cohomologies agree, compatibly with Tate twists?