Locally free not implies free

Statement
It is possible to have a locally free group that is not a free group, i.e., every finitely generated subgroup of the group is free but the whole group is not free.

Abelian examples
These are abelian groups that are locally cyclic aperiodic groups that are not isomorphic to the group of integers. The simplest example is $$\mathbb{Q}$$, the additive group of rational numbers. For a complete classification, refer to classification of locally cyclic aperiodic groups.

Non-abelian examples
One way to build non-abelian examples from the abelian ones is using free products. Thus, the free product of two copies of the group of rational numbers is a locally free group that is not free.