Back in 1983, Gupta and Said (Math Z.) were constructing finitely generated infinite p-groups as groups of automorphisms of p-regular trees. There is a later treatment of their work by Gilbert Baumslag, but I don't have an immediate reference for his exposition. I think that Baumslag mostly looks at the special case where p=3. This work might be irrelevant to Dominik, but it might be of interest.
All trees are planar.