To move this off the unanswered queue, the answer is yes. For any $\mathcal{M}\models\mathsf{PA}$, the set $\mathcal{M}_{\mathit{def}}$ of parameter-freely-definable elements of $\mathcal{M}$ is an elementary substructure of $\mathcal{M}$ by the Tarski-Vaught test. (More generally this occurs in any theory with definable Skolem functions, another important example being $\mathsf{ZFC+V=L}$.)

This gives us the following quite nice result:

$Th(-)$ provides a bijection between isomorphism types of pointwise-definable models of $\mathsf{PA}$ and completions of $\mathsf{PA}$.

That is, every completion $T$ of $\mathsf{PA}$ has exactly one pointwise-definable model up to isomorphism, and this model is nonstandard iff $T\not=\mathsf{TA}$. A key step to proving the above is the following:

If $\mathcal{A},\mathcal{B}$ are elementarily equivalent and each is pointwise-definable, then $\mathcal{A}\cong\mathcal{B}$.