NP-hard problem on planar unit disk graph I am curious to know whether there are problems which are np-hard even on planar unit disk graphs.

A unit disk graph is the intersection graph of a collection of unit disks in the plane, where we draw a vertex for each unit disk. 

Maybe some problem on a accordingly restricted grid graph?

alex
Did you try doing a search for "planar unit disk graphs"? According to this paper the problem of finding the largest possible induced matching is NP-complete for planar unit disk graphs, according to this paper so is the minimum connected dominating set, and according to this paper so is the maximum number of disjoint dominating sets.