Does every countable, infinite, amenable, periodic group $G$ contain an infinite locally finite subgroup?
I would be happy with an infinitely generated counterexample as long as it is countable.
A counterexample cannot be elementary amenable, as elementary amenable periodic groups are locally finite.
First Grigorchuk's group is not a counterexample: it is a 2-group, and hence it contains an infinite abelian group (by a result of Held, "On abelian subgroups of infinite 2-groups").
Amenability of $G$ is essential, due to existence of Tarski monsters whose subgroups are finite cyclic (constructed by Olshanskii).
Any infinite locally finite group contains an infinite abelian subgroup (see a paper of Hall-Kulatilaka here, so it is equivalent to ask whether $G$ contains an infinite abelian subgroup.
This question was asked here in 2008, so perhaps it is an open problem.