Hodge theory for non-Archimedean analytic spaces

By Deligne's Hodge theory, the integral cohomology groups $H^n(\mathcal{X}^h,\mathbf{Z})$ of the $\mathbf{C}$-analytification $\mathcal{X}^h$ of a separated scheme $\mathcal{X}$ of finite type over $\mathbf{C}$ are provided with a mixed Hodge structure, functorial in $\mathcal{X}$. Given a non-Archimedean field $K$ isomorphic to the field of Laurent power series $\mathbf{C}((z))$, there is a functor $\mathcal{X}\mapsto \mathcal{X}^{\text{an}}_K$ that takes $\mathcal{X}$ to the non-Archimedean $K$-analytification of $\mathcal{X}_K = \mathcal{X}\otimes_{\mathbf{C}} K$. I'll talk about the extension of Deligne's Hodge theory via the above functor to a full subcategory of the category of $K$-analytic spaces that contains the $K$-analytifications of schemes of finite type over $K$, proper $K$-analytic spaces, and the analytic subdomains of both.