As is well known, the group $\mathrm{PSL}(2,\mathbf{Z})$ is isomorphic to the free product of two cyclic groups of orders 2 and 3.

Is there a similar description of the projective special linear group over p-adic integers? If yes, where can I find it? If no, what is know on the algebraic structure of $\mathrm{PSL}(2,\mathbf{Z}_p)$?