A closely related result is proven in [Th07, 3.26]. We will explain how our statement can be deduced from that result. Following the notation in [Th07], we denote by $\mathscr{X}^{\beth}$ the $k$-analytic space associated to the toroidal embedding $X\hookrightarrow\mathscr{X}$, where $k$ is endowed with the trivial absolute value. By definition, $\mathscr{X}^{\beth}$ is the generic fiber of the formal $t$-adic completion $\widehat{\mathscr{X}}$ of $\mathscr{X}$, viewed as a special formal $k$-scheme by forgetting the $k[\negthinspace[t]\negthinspace]$-structure [Be96, §1].

The relation between $\mathscr{X}^{\beth}$ and $X_{K}^{\mathrm{an}}$ is explained in detail at the beginning of Section 4 in [Ni11]; let us recall the main idea. Considering the morphism of special formal $k$-schemes $\widehat{\mathscr{X}}\to\mathrm{Spf}\,k[\negthinspace[t]\negthinspace]$ and passing to the generic fibers, we obtain a morphism of $k$-analytic spaces from $\mathscr{X}^{\beth}$ to the open unit disc $D$ over $k$. We can identify the underlying topological space of $D$ with $[0,1[$ by means of the homeomorphism

$$D\to[0,1[\,:x\mapsto|t(x)|.$$

The residue field of $D$ at the point $1/e$ in $[0,1[$ is $K$ with our chosen $t$-adic absolute value $|\cdot|_{K}$, and the $K$-analytic space $X_{K}^{\mathrm{an}}$ is canonically isomorphic to the fiber of $\mathscr{X}^{\beth}$ over $1/e$. Thus we can view $X_{K}^{\mathrm{an}}$ as the subspace of $\mathscr{X}^{\beth}$ consisting of the points $x$ such that $|t(x)|=1/e$.

In [Th07, 3.13], Thuillier constructs a retraction $p_{\mathscr{X}}$ of $\mathscr{X}^{\beth}$ onto a certain subspace $\mathcal{S}(\mathscr{X})$, the skeleton of the toroidal embedding. Moreover, in [Th07, 3.26], he shows that $p_{\mathscr{X}}$ can be extended to a strong deformation retraction $H$ of $\mathscr{X}^{\beth}$ onto $\mathcal{S}(\mathscr{X})$. Going through the definitions, one observes that $p_{\mathscr{X}}$ and $H$ commute with the morphism $\mathscr{X}^{\beth}\to D$ and that the restriction of

$$p_{\mathscr{X}}:\mathscr{X}^{\beth}\to\mathcal{S}(\mathscr{X})$$

over the point $1/e$ of $D$ is precisely the retraction

$$\rho_{\mathscr{X}}:X_{K}^{\mathrm{an}}\to\mathrm{Sk}(\mathscr{X}).$$

Thus by restricting $H$ over $1/e\in D$, we obtain a map that satisfies all the properties in the statement. ∎