The usual crystalline comparison theorem involves a comparison of cohomology between the generic and special fibre of a proper smooth scheme over a $p$-adic discrete valuation ring. To formulate an analog where the generic fibre is merely a rigid space, one option is to use formal models, as in the following result.
Theorem: Fix a finite extension $K/\mathbf{Q}_p$. Say $\mathfrak{X}/\mathcal{O}_K$ is a proper smooth proper formal scheme whose generic fibre (in the sense of rigid analytic geometry) is the proper smooth rigid space $X/K$. Then there is a crystalline comparison isomorphism relating the crystalline cohomology of the special fibre $\mathfrak{X}_0$ of $\mathfrak{X}/\mathcal{O}_K$ with the 'etale cohomology of the geometric generic fibre $X_{\overline{K}}$.
One reference is the paper by Colmez-Niziol (https://arxiv.org/abs/1505.06471). This paper proves a general comparison result for semistable formal schemes, but uses a different method than Scholze's paper.
The paper by Bhatt-Morrow-Scholze (https://arxiv.org/abs/1602.03148) also includes a generalization of the above theorem for formal schemes that are merely defined over the valuation ring of algebraically closed field. This proof uses methods closer to the Faltings-Scholze approach, but with integral enhancements.