The group $G$ described in " Simple(st) example of an infinite $p$-group with trivial center " is in fact an ICC group. This means that every non-trivial element $g\in G$ has an infinite conjugacy class. As a consequence, $G$ has trivial center (because the conjugacy class of an element in the center contains only one element), and no finite normal subgroups (because every normal subgroup is a union of conjugacy classes).

The group $\mathbb{Z}/p\wr(\mathbb{Z}/p)^\infty$ is also an ICC (infinite conjugacy classes) p-group, so its center is trivial and it can not have finite normal subgroups.