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Added. Since the question concerned an arbitrary hermitian metric, I felt I should add a pointer to Chambert-Loir and Ducros's work on Chern forms and their associated measures: Formes differentielles reelles et courants sur les espaces de Berkovich. Arakelov geometry considers line bundles $\bar{L} = (L,\| \cdot\|)$ with a smooth hermitian metric. The Chern (curvature) form $c_1( \bar{L})$ is then a smooth $(1,1)$-form, and its determinant (or self intersection) is either zero or a full support signed measure on $X(\mathbb{C}) = X_{/\mathbb{C}}^{\mathrm{an}}$. For a $p$-adic metric $\|\cdot\|_p$ in $L$ there are a corresponding form and measure on $X_{/\mathbb{C}_p}^{\mathrm{an}}$. But if this metric arises from an integral model $(\mathfrak{X},\mathfrak{L})$ over $\mathbb{Z}_p$, where $\mathfrak{X}_{/\mathbb{F}_p}$ is regular, then the obtained (signed) measure charges a single point of the Berkovich space: the unique point reducing to the generic point of the scheme $\mathfrak{X}_{/\mathbb{F}_p}$. This is nothing like the Archimedean case with a $C^{\infty}$ metric; to get the measures of full dimensional support $X$ needs to be maximally degenerate at $p$.

Added. Since the question concerned an arbitrary hermitian metric, I felt I should add a pointer to Chambert-Loir and Ducros's work on Chern forms and their associated measures: Formes differentielles reelles et courants sur les espaces de Berkovich. Arakelov geometry considers line bundles $\bar{L} = (L,\| \cdot\|)$ with a smooth hermitian metric. The Chern (curvature) form $c_1( \bar{L})$ is then a smooth $(1,1)$-form, and its determinant (or self intersection) is either zero or a full support signed measure on $X(\mathbb{C}) = X_{/\mathbb{C}}^{\mathrm{an}}$. For a $p$-adic metric $\|\cdot\|_p$ in $L$ there are a corresponding form and measure on $X_{/\mathbb{C}_p}^{\mathrm{an}}$. But if this metric arises from an integral model $(\mathfrak{X},\mathfrak{L})$ over $\mathbb{Z}_p$, where $\mathfrak{X}_{/\mathbb{F}_p}$ is regular, then the obtained (signed) measure charges a single point of the Berkovich space: the unique point reducing to the generic point of the scheme $\mathfrak{X}_{/\mathbb{F}_p}$.

Added. Since the question concerned an arbitrary hermitian metric, I felt I should add a pointer to Chambert-Loir and Ducros's work on Chern forms and their associated measures: Formes differentielles reelles et courants sur les espaces de Berkovich. Arakelov geometry considers line bundles $\bar{L} = (L,\| \cdot\|)$ with a smooth hermitian metric. The Chern (curvature) form $c_1( \bar{L})$ is then a smooth $(1,1)$-form, and its determinant (or self intersection) is either zero or a full support signed measure on $X(\mathbb{C}) = X_{/\mathbb{C}}^{\mathrm{an}}$. There are a corresponding form and measure on $X_{/\mathbb{C}_p}^{\mathrm{an}}$, where the $p$-adic metric in $L$ is taken to arise from the integral model of $L$ over $\mathbb{Z}_p$. Now if $X$ has a good reduction at $p$, the obtained (signed) measure charges a single point of the Berkovich space. This is nothing like the Archimedean case with a $C^{\infty}$ metric; to get the measures of full dimensional support $X$ needs to be maximally degenerate at $p$.

Added. Since the question concerned an arbitrary hermitian metric, I felt I should add a pointer to Chambert-Loir and Ducros's work on Chern forms and their associated measures: Formes differentielles reelles et courants sur les espaces de Berkovich. Arakelov geometry considers line bundles $\bar{L} = (L,\| \cdot\|)$ with a smooth hermitian metric. The Chern (curvature) form $c_1( \bar{L})$ is then a smooth $(1,1)$-form, and its determinant (or self intersection) is either zero or a full support signed measure on $X(\mathbb{C}) = X_{/\mathbb{C}}^{\mathrm{an}}$. For a $p$-adic metric $\|\cdot\|_p$ in $L$ there are a corresponding form and measure on $X_{/\mathbb{C}_p}^{\mathrm{an}}$. But if this metric arises from an integral model $(\mathfrak{X},\mathfrak{L})$ over $\mathbb{Z}_p$, where $\mathfrak{X}_{/\mathbb{F}_p}$ is regular, then the obtained (signed) measure charges a single point of the Berkovich space: the unique point reducing to the generic point of the scheme $\mathfrak{X}_{/\mathbb{F}_p}$. This is nothing like the Archimedean case with a $C^{\infty}$ metric; to get the measures of full dimensional support $X$ needs to be maximally degenerate at $p$.

Added. Since the question concerned an arbitrary hermitian metric, I felt I should add a pointer to Chambert-Loir and Ducros's work on Chern forms and their associated measures: Formes differentielles reelles et courants sur les espaces de Berkovich. Arakelov geometry considers line bundles $\bar{L} = (L,\| \cdot\|)$ with a smooth hermitian metric. The Chern (curvature) form $c_1( \bar{L})$ is then a smooth $(1,1)$-form, and its determinant (or self intersection) is a full support measure on $X(\mathbb{C}) = X_{/\mathbb{C}}^{\mathrm{an}}$. There are a corresponding form and measure on $X_{/\mathbb{C}_p}^{\mathrm{an}}$, where the $p$-adic metric in $L$ is taken to arise from the integral model of $L$ over $\mathbb{Z}_p$. Now if $X$ has a good reduction at $p$, the obtained measure is the Dirac mass charging a single point of the Berkovich space. This is nothing like the Archimedean case with a $C^{\infty}$ metric, but if $X$ has a sufficiently bad reduction at $p$ then the measure may likewise have a full dimensional support.

Added. Since the question concerned an arbitrary hermitian metric, I felt I should add a pointer to Chambert-Loir and Ducros's work on Chern forms and their associated measures: Formes differentielles reelles et courants sur les espaces de Berkovich. Arakelov geometry considers line bundles $\bar{L} = (L,\| \cdot\|)$ with a smooth hermitian metric. The Chern (curvature) form $c_1( \bar{L})$ is then a smooth $(1,1)$-form, and its determinant (or self intersection) is either zero or a full support signed measure on $X(\mathbb{C}) = X_{/\mathbb{C}}^{\mathrm{an}}$. There are a corresponding form and measure on $X_{/\mathbb{C}_p}^{\mathrm{an}}$, where the $p$-adic metric in $L$ is taken to arise from the integral model of $L$ over $\mathbb{Z}_p$. Now if $X$ has a good reduction at $p$, the obtained (signed) measure charges a single point of the Berkovich space. This is nothing like the Archimedean case with a $C^{\infty}$ metric; to get the measures of full dimensional support $X$ needs to be maximally degenerate at $p$.