I recently found some references: Theorem 4.5 of this paper and Theorem 4 + next Corollary of this paper which says:
If $(R,m,k)$ is a complete normal local domain of dimension $2$ such that $k$ is the algebraic closure of some finite field, then $Cl(R)$ is torsion!
It remains open what happens if $k$ is not algebraically closed.