By compactness of $U\setminus U^{\circ}$, there exists a compact strictly $K$-analytic domain $Z\subset V$ such that $Z$ is a neighbourhood of $U\setminus U^{\circ}$ and $W\cap Z=\emptyset$. Hence $W\coprod Z$ is a compact strictly $K$-analytic domain of $V$ and we consider the piecewise linear function on $W\coprod Z$ defined by $f$ on $W$ and by $0$ on $Z$. Then we apply Proposition 2.6 to $L=\mathcal{O}_{V}$, in which case formal metrics correspond to piecewise linear functions (see Proposition 2.8). We deduce that there exists a piecewise linear function $g\colon V\to{\mathbb{R}}$ which agrees with $f$ on $W$ and which agrees with $0$ on $Z$. But since $Z$ is a neighborhood of $U\setminus U^{\circ}$, we deduce that the function $\varphi\colon V\to{\mathbb{R}}$ defined by

$$\varphi(x)=\begin{cases}g(x)&{\rm if}\ x\in U\\ 0&{\rm if}\ x\notin U\end{cases}$$

is still piecewise linear. Since $\varphi$ extends $f$ and ${\rm supp}(\varphi)\subset U$, we get the claim. ∎