Lackenby ( http://people.maths.ox.ac.uk/lackenby/lg070105.ps ) proved that a finitely presented group of positive rank gradient is large, i.e. it contains a subgroup of finite index that projects onto a non-abelian free group. So while the answer to the original question is no, for finitely presented groups it becomes yes in a very strong sense.

Lackenby ( http://people.maths.ox.ac.uk/lackenby/lg070105.ps ) proved that a finitely presented group, which has a pro-$p$ completion of positive rank gradient, is large, i.e. it contains a subgroup of finite index that projects onto a non-abelian free group. So while the answer to the original question is no, if you add some extra conditions it becomes yes in a very strong sense.