Is there any example of group $G$ satisfying the following properties?
- $G$ is non-abelian, infinitely generated (i.e. it is not finitely generated) and not a free group.
- $H< G$ implies that $H$ is a free group.
Clearly such an example should be torsion-free and countable.
(Added, from comments) For completeness, here's why it has to be countable. First, it is enough to construct a subgroup of index 2. Indeed, a theorem of Swan asserts that a torsion-free group with a free finite-index subgroup is free. Next, if a group $G$ has no subgroup of index 2 (i.e. every element is product of squares), it is easy to construct a nontrivial countable subgroup $H$ with the same property). In the current setting, $G$ is uncountable so $H$ is a proper subgroup, and hence has to be free, contradiction.