Abelian subgroup in an infinite non-abelian 3-group

Does an infinite non-abelian 3-group of exponent greater than or equal to 9 have an infinite abelian subgroup?

I know that every 2-group or 3-group of exponent 3 has an infinite abelian subgroup. I wonder whether the result holds or not for 3-groups of higher exponent.

• In the Burnside group with two generators and exponent $3^n$, for sufficiently large $n$, every abelian group is cyclic, hence finite. This is pointed out in the answer to the same question in math.stackexchange.com/questions/496086/… – Ashot Minasyan Nov 14 '13 at 10:03
• The existence of an infinite abelian subgroup is shared by all infinite locally finite groups. For infinite finitely generated torsion groups, it's not always true, as given by Ashot's comment. – YCor Jul 21 at 10:55

(cw answer, copied from Ashot Minasyan's comment.) In the Burnside group with two generators and exponent $$3^n$$, for sufficiently large $$n$$, every abelian group is cyclic, hence finite. This is pointed out in the answer to the same question in this MathSE post.