Would you please construct a locally nilpotent group which is not nilpotent? Would you please construct a locally nilpotent group which is not nilpotent?
An example of a locally nilpotent group which is not nilpotent?

The Fitting subgroup $\mathrm{Fit}(G)$ of a group $G$ is the product of all nilpotent normal subgroups of $G$. Since the product of finitely many nilpotent normal subgroups is again a nilpotent normal subgroup, $\mathrm{Fit}(G)$ is locally nilpotent. To find your example it therefore suffices to find a nonnilpotent group with $G=\mathrm{Fit}(G)$.
In fact, if $p$ is a prime and $E$ is an infinite elementary abelian $p$-group (i.e. $E$ is abelian of exponent $p$), then the wreath product $\mathbb{Z}_p\wr E$ satisfies this property, but the proof (and even the construction of the wreath product, if you haven't seen it before) is somewhat involved.
