[Definitions: The socle of a group $G$ is the subgroup generated by the minimal normal subgroups of $G$. The term "rank" is meant in the sense of the Mal’cev special or Prüfer rank: by definition a group $H$ has finite rank $\le n$ if every finitely generated subgroup of $X$ can be generated by $\le n$ elements. A group which does not have finite rank is said to have infinite rank. Below $p$ denotes a given prime number.]

The socle of an abelian $p$-group of infinite rank has infinite rank. I wonder if it is true in the locally nilpotent case:

Is the following assertion true:

Let $G$ be locally nilpotent $p$-group. If $G$ has infinite rank, then the socle of $G$ has infinite rank.