Clearly, every formal metric is piecewise linear. To prove the converse, we may assume that $V$ is connected. It is a general fact from topology (see [Bou71, chap. 1, §9, Théorème 5]) that a connected locally compact space is paracompact if and only if it is countable at infinity. It follows that there is a finite or a countable ${\rm G}$-open covering $(V_{i})_{i\in I}$ of $V$ of finite type by strictly $K$-affinoid domains $V_{i}$ with frames $s_{i}$ of $L|_{V_{i}}$ such that $\|s_{i}\|=1$ on $V_{i}$. Then $V_{i}$ is the Berkovich spectrum of a strictly $K$-affinoid algebra ${\mathscr{A}}_{i}$. Obviously, there is an admissible ${K^{\circ}}$-algebra $A_{i}$ with ${\mathscr{A}}_{i}=A_{i}\otimes_{K^{\circ}}K$. For $f_{i}\in{\mathscr{A}}_{i}^{\circ}$, the ${K^{\circ}}$-algebra $A_{i}[f_{i}]$ is

an admissible ${K^{\circ}}$-algebra [Bos14, Lemma 8.4.6].

Using the existence of a formal metric on $L^{\rm an}$, we may assume that $L={\mathcal{O}}_{V}$ and hence the frames $s_{i}$ are invertible functions on the sets $V_{i}$. Using that $V$ is paracompact, the underlying rigid space is quasiseparated and hence $V_{ij}=V_{i}\cap V_{j}={\rm Spf}({\mathscr{A}}_{ij})$ for some strictly $K$-affinoid algebra ${\mathscr{A}}_{ij}$. If $f_{ij}=s_{i}/s_{j}$, then $f_{ij}\in({\mathscr{A}}_{ij}^{\circ})^{\times}$. Using the above, we choose a formal affine ${K^{\circ}}$-model ${\mathfrak{W}}_{ij}$ with generic fiber $V_{ij}$ such that $f_{ij}\in{\mathcal{O}}({\mathfrak{W}}_{ij})$.

In the following, we assume that $I={\mathbb{N}}\setminus\{0\}$ (the finite case is similar and even easier) and we consider $k\in{\mathbb{N}}$. By an inductive procedure, we will construct a formal model ${\mathfrak{V}}^{(k)}$ of $V$ such that $V_{i}$ is the generic fiber of a formal open subset ${\mathfrak{V}}_{i}^{(k)}$ of ${\mathfrak{V}}^{(k)}$ for every $i\in I$ and such that ${\mathfrak{V}}_{i}^{(k)}\cap{\mathfrak{V}}_{j}^{(k)}$ is lying over ${\mathfrak{W}}_{ij}$ for every $i,j\in\{0,\dots,k\}$. By this we mean that for every $i,j\in\{0,\dots,k\}$ there exists a morphism ${\mathfrak{V}}_{i}^{(k)}\cap{\mathfrak{V}}_{j}^{(k)}\to{\mathfrak{W}}_{ij}$ which is the identity on the generic fibre.

Note that the case $k=0$ follows from [Bos14, Lemma 8.4.5]. Let $k\geq 1$ and assume that ${\mathfrak{V}}^{(k-1)}$ is already constructed. By Raynaud’s theorem and [BL93b, Corollary 5.4], there is an admissible formal blowing up $p_{k}$ of ${\mathfrak{V}}_{k}^{(k-1)}$ such that $V_{ik}$ (resp. $V_{ki}$) is the generic fiber of a formal open subset lying over ${\mathfrak{W}}_{ik}$ (resp. ${\mathfrak{W}}_{ki}$) for $i=1,\dots,k$. By [Bos14, Proposition 8.2.13], we may extend $p_{k}$ to an admissible formal blowing up ${\mathfrak{V}}^{(k)}$ of ${\mathfrak{V}}^{(k-1)}$ with center $Z$ in the special fiber such that $Z$ is disjoint from every ${\mathfrak{V}}_{i}^{(k-1)}$ with $i\leq k-1$ satisfying ${\mathfrak{V}}_{i}^{(k-1)}\cap{\mathfrak{V}}_{k}^{(k-1)}=\emptyset$. Then ${\mathfrak{V}}^{(k)}$ satisfies the claim with ${\mathfrak{V}}_{i}^{(k)}$ equal to the preimage of ${\mathfrak{V}}_{i}^{(k-1)}$ in ${\mathfrak{V}}^{(k)}$.

Using that the $\rm G$-covering $(V_{i})_{i\in I}$ is of finite type, the above construction shows that the formal models ${\mathfrak{V}}^{(k)}$ eventually become stable over ${\mathfrak{V}}_{i}^{(0)}$ for any $i\in I$ and hence we get a formal model ${\mathfrak{V}}$ of $V$ lying above all the models ${\mathfrak{V}}^{(k)}$. It has the property that every $V_{i}$ is the generic fiber of a formal open subset ${\mathfrak{V}}_{i}$ and that ${\mathfrak{V}}_{i}\cap{\mathfrak{V}}_{j}$ is lying over ${\mathfrak{W}}_{ij}$ for every $i,j\in I$. Since $f_{ij}$ and $f_{ji}$ are both in ${\mathcal{O}}({\mathfrak{V}}_{i}\cap{\mathfrak{V}}_{j})$, we see that $f_{ij}$ is invertible on ${\mathfrak{V}}_{i}\cap{\mathfrak{V}}_{j}$. This means that $(s_{i})_{i\in I}$ is a vertical Cartier divisor on ${\mathfrak{V}}$ inducing the metric. ∎