A finite $p$-group $G$ always has a non-trivial center. However, there exist infinite $p$-groups with trivial center, see for example this question Simple(st) example of an infinite $p$-group with trivial center. My question is do there exist such infinite $p$-groups with trivial center that contain no finite normal subgroups? More generally, does there exist an infinite $p$-group $G$ such that infinitely many elements of $G$ are not in $Z(G)$ and such that $G$ has no finite normal subgroups?