Locally nilpotent groups

$p$-groups are locally nilpotent groups? How can we show it? I know that a finite $p$-group is nilpotent. If in general the answer is "no", then what about a FC $p$-group?

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Tarski monsters are non-(locally nilpotent) p-groups. – Jack Schmidt Oct 12 '12 at 16:45
There are finitely generated p groups which are not nilpotent. – Mustafa Gokhan Benli Oct 12 '12 at 16:48
Thanks. Do You know something about $FC\space p$-groups? Are they locally nilpotent groups? – W4cc0 Oct 12 '12 at 16:48

In periodic $FC$-groups Dietzmann's lemma forces $G$ to be locally nipotent.

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