  We analytically derive a class of non-singular, static and spherically
symmetric topological black hole metrics inF(R)-gravity. These have not a de
Sitter core at their centre, as most model in standard General Relativity. We
study the geometric properties and the motion of test particles around these
objects. Since they have two horizons, the inner being of Cauchy type, we focus
on the problem of mass inflation and show that it occurs except when some
extremal conditions are met.
