Let $\mathrm{X}$ be a $K$-analytic space in the sense of Berkovich [11]. For simplicity, we will assume that $\mathrm{X}$ is the analytic space associated to a proper scheme over $K$. In that context, the general definition of continuous metrized line bundles given above makes sense.

Let us detail the example of the line bundle $\mathscr{O}(1)$ on the projective space $\mathrm{P}^{n}_{K}$. A point $x\in\mathrm{P}^{n}_{K}$ possesses a complete residue field $\mathscr{H}(x)$ which is a complete extension of $K$ and homogeneous coordinates $[x_{0}:\dots:x_{n}]$ in the field $\mathscr{H}(x)$. As in complex geometry, the projective space $\mathrm{P}^{n}_{K}$ is obtained by glueing $n+1$ copies $\mathrm{U}_{0},\dots,\mathrm{U}_{n}$ of the affine space $\mathrm{A}^{n}_{K}$, where $\mathrm{U}_{i}$ corresponds to those points $x$ such that $x_{i}\neq 0$. Recall also that $\mathrm{A}^{n}_{K}$ is the space of multiplicative semi-norms on the $K$-algebra $K[T_{1},\dots,T_{n}]$ which induce the given absolute value on $K$, together with the coarsest topology such that for any semi-norm $x\in\mathrm{A}^{n}_{K}$, the map $K[T_{1},\dots,T_{n}]\rightarrow{\mathbf{R}}$ defined by $f\mapsto x(f)$ is continuous. The kernel of a semi-norm $x$ is a prime ideal $\mathfrak{p}_{x}$ of $K[T_{1},\dots,T_{n}]$ and $x$ induces a norm on the quotient ring $K[T_{1},\dots,T_{n}]/\mathfrak{p}_{x}$, hence on its field of fractions $K(x)$. The completion of $K(x)$ with respect to this norm is denoted $\mathscr{H}(x)$ and is called the complete residue field of $x$. The images in $\mathscr{H}(x)$ of the intederminates $T_{i}$ are denoted $T_{i}(x)$, more generally, the image in $\mathscr{H}(x)$ of any polynomial $f\in K[T_{1},\dots,T_{n}]$ is denoted $f(x)$ ; one has $x(f)=\left|{f(x)}\right|$.

Let $f$ be a rational function on $\mathrm{P}^{n}_{K}$, that is an element of $\in K(T_{1},\dots,T_{n})$. It defines an actual function on the open set $\mathrm{U}$ of ${\mathbf{P}}^{n}_{K}$ where its denominator does not vanish ; its value at a point $x\in U$ is an element of $\mathscr{H}(x)$. More generally, Berkovich defines an analytic function on an open set $\mathrm{U}$ of $\mathrm{P}^{n}_{K}$ as a function $f$ on $U$ such that $f(x)\in\mathscr{H}(x)$ for any $x\in\mathrm{U}$, and such that any point $x\in\mathrm{U}$ possesses a neighbourhood $\mathrm{V}\subset\mathrm{U}$ such that $f|_{\mathrm{V}}$ is a uniform limit of rational functions without poles on $\mathrm{V}$.

The line bundle $\mathscr{O}(1)$ can also be defined in a similar way to the classical case ; by a similar GAGA theorem, its global sections are exactly the same as in algebraic geometry and are described by homogeneous polynomials of degree $1$ with coefficients in $K$. If $P$ is such a polynomial and $s_{P}$ the corresponding section, then

$$\left\|{s_{P}}\right\|(x)=\frac{\left|{P(x_{0},\dots,x_{n})}\right|}{\max(\left|{x_{0}}\right|,\dots,\left|{x_{n}}\right|)}$$

where $[x_{0}:\dots:x_{n}]$ is a system of homogeneous coordinates in $\mathscr{H}(x)$ for the point $x$. The function $\left\|{s_{P}}\right\|$ is continuous on $\mathrm{P}^{n}_{K}$, by the very definition of the topology on $\mathrm{P}^{n}_{K}$. Using the fact that $\mathscr{O}(1)$ is generated by its global sections, one deduces the existence of a continuous metric on $\mathscr{O}(1)$ satisfying the previous formula.