As requested in the comments, here's an example of a local, normal $2$-dimensional domain R whose class group in positive characteristic such that $\mathrm{Cl}(R)$ is not torsion: choose an elliptic curve $E \subset \mathbf{P}^2$ over a field $k$ such that $E(k)$ is not torsion, and take R to be the localisation local ring at the origin of the affine cone on $E$ (i.e., $R = k[x,y,z]/(f)$ k[x,y,z]/(f)_{(x,y,z)}$where$f$is a homoegenous cubic defining$E$). This can be done over$k = \overline{\mathbf{F}_p(t)}$. Proof: The normality follows from the fact that R is a hypersurface singularity (hence even Gorenstein) and isolated and$2$-dimensional (hence regular in codim 1). Blowing up at the origin defines a map$f:X \to \mathrm{Spec}(R)$. One can then show the following:$X$is smooth, and$X$can be identified with the Zariski localisation along the zero section of the total space of the line bundle$L = \mathcal{O}_{\mathbf{P^2}}(-1)|_E$(these are general facts about cones). By Lipman's theorem, it suffices to show that$\mathrm{Pic}^0(X)$contains non-torsion elements. As$X$is fibered over$E$with a section, the pullback$\mathrm{Pic}^0(E) \to \mathrm{Pic}^0(X)$is a direct summand. As$\mathrm{Pic}^0(E) \simeq E(k)$has non-torsion elements by assumption, so does$\mathrm{Pic}^0(X)$\mathrm{Pic}^0(X)$.
Also, an additional comment: In general, Lipman's theorem tells you that $\mathrm{Cl}(R)$ is torsion if and only if $\mathrm{Pic}^0(X)$ is torsion. Now $\mathrm{Pic}(X) \simeq \lim_n \mathrm{Pic}(X_n)$ where $X_n$ is the $n$-th order thickening of the exceptional fibre $E$. Because we are blowing up a point, the sheaf of ideals $I$ defining $E$ is ample on $E$. The kernel and cokernel of $\mathrm{Pic}(X_n) \to \mathrm{Pic}(X_{n-1})$ are identified with $H^1(E,I|_E^{\otimes n+1})$ and $H^2(E,I|_E^{\otimes n+1})$. As $I|_E$ is ample, it follows that the system "$\lim_n \mathrm{Pic}(X_n)$" is eventually stable. Thus, $\mathrm{Pic}(X) \simeq \mathrm{Pic}(X_n)$ for $n$ sufficiently big. As $X_n$ is a proper variety, it follows that if we are working over a finite field (resp. an algebraic closure of a finite field), then $\mathrm{Pic}^0(X)$ is finite (resp. ind-finite).
As requested in the comments, here's an example of a local, normal domain R whose class group is not torsion: choose an elliptic curve $E \subset \mathbf{P}^2$ over a field $k$ such that $E(k)$ is not torsion, and take R to be the localisation at the origin of the affine cone on $E$ (i.e., $R = k[x,y,z]/(f)$ where $f$ is a homoegenous cubic defining $E$). This can be done over $k = \overline{\mathbf{F}_p(t)}$.
Proof: Blowing up at the origin defines a map $f:X \to \mathrm{Spec}(R)$. One can then show the following: $X$ is smooth, and $X$ can be identified with the Zariski localisation along the zero section of the total space of the line bundle $L = \mathcal{O}_{\mathbf{P^2}}(-1)|_E$ (these are general facts about cones). By Lipman's theorem, it suffices to show that $\mathrm{Pic}^0(X)$ contains non-torsion elements. As $X$ is fibered over $E$ with a section, the pullback $\mathrm{Pic}^0(E) \to \mathrm{Pic}^0(X)$ is a direct summand. As $\mathrm{Pic}^0(E) \simeq E(k)$ has non-torsion elements by assumption, so does $\mathrm{Pic}^0(X)$
Also, an additional comment: In general, Lipman's theorem tells you that $\mathrm{Cl}(R)$ is torsion if and only if $\mathrm{Pic}^0(X)$ is torsion. Now $\mathrm{Pic}(X) \simeq \lim_n \mathrm{Pic}(X_n)$ where $X_n$ is the $n$-th order thickening of the exceptional fibre $E$. Because we are blowing up a point, the sheaf of ideals $I$ defining $E$ is ample on $E$. The kernel and cokernel of $\mathrm{Pic}(X_n) \to \mathrm{Pic}(X_{n-1})$ are identified with $H^1(E,I^{n+1}/I^n)$ H^1(E,I|_E^{\otimes n+1})$and$H^2(E,I^{n+1}/I^n)$. H^2(E,I|_E^{\otimes n+1})$. As $I$ I|_E$is ample, it follows that the system "$\lim_n \mathrm{Pic}(X_n)$" is eventually stable. Thus,$\mathrm{Pic}(X) \simeq \mathrm{Pic}(X_n)$for$n$sufficiently big. As$X_n$is a proper curvevariety, it follows that if we are working over a finite field (resp. an algebraic closure of a finite field), then$\mathrm{Pic}^0(X)$is finite (resp. ind-finite). 1 As requested in the comments, here's an example of a local, normal domain R whose class group is not torsion: choose an elliptic curve$E \subset \mathbf{P}^2$over a field$k$such that$E(k)$is not torsion, and take R to be the localisation at the origin of the affine cone on$E$(i.e.,$R = k[x,y,z]/(f)$where$f$is a homoegenous cubic defining$E$). This can be done over$k = \overline{\mathbf{F}_p(t)}$. Proof: Blowing up at the origin defines a map$f:X \to \mathrm{Spec}(R)$. One can then show the following:$X$is smooth, and$X$can be identified with the Zariski localisation along the zero section of the total space of the line bundle$L = \mathcal{O}_{\mathbf{P^2}}(-1)|_E$(these are general facts about cones). By Lipman's theorem, it suffices to show that$\mathrm{Pic}^0(X)$contains non-torsion elements. As$X$is fibered over$E$with a section, the pullback$\mathrm{Pic}^0(E) \to \mathrm{Pic}^0(X)$is a direct summand. As$\mathrm{Pic}^0(E) \simeq E(k)$has non-torsion elements by assumption, so does$\mathrm{Pic}^0(X)$Also, an additional comment: In general, Lipman's theorem tells you that$\mathrm{Cl}(R)$is torsion if and only if$\mathrm{Pic}^0(X)$is torsion. Now$\mathrm{Pic}(X) \simeq \lim_n \mathrm{Pic}(X_n)$where$X_n$is the$n$-th order thickening of the exceptional fibre$E$. Because we are blowing up a point, the sheaf of ideals$I$defining$E$is ample on$E$. The kernel and cokernel of$\mathrm{Pic}(X_n) \to \mathrm{Pic}(X_{n-1})$are identified with$H^1(E,I^{n+1}/I^n)$and$H^2(E,I^{n+1}/I^n)$. As$I$is ample, it follows that the system "$\lim_n \mathrm{Pic}(X_n)$" is eventually stable. Thus,$\mathrm{Pic}(X) \simeq \mathrm{Pic}(X_n)$for$n$sufficiently big. As$X_n$is a proper curve, it follows that if we are working over a finite field (resp. an algebraic closure of a finite field), then$\mathrm{Pic}^0(X)\$ is finite (resp. ind-finite).