TITLE: Is there a perfect set of ground model reals in the Cohen extension?
QUESTION [6 upvotes]: This question is motivated by the "interesting tidbit" in Hamkins' response here: https://mathoverflow.net/a/99025/10671, in which he demonstrates that, after Cohen forcing, there is a perfect set consisting entirely of mutually generic Cohen reals.  In particular, this demonstrates that there is a perfect set in the extension containing no ground model reals.
I'm interested in a related question and, while I feel it must have been answered, I can't find a reference.

QUESTION. Consider the reals in the extension $V[c]$ obtained by forcing to add
  a Cohen real.  Does there exist a perfect set in $V[c]$ consisting entirely of ground model reals? Or does every perfect set of reals in $V[c]$
  contain a "new" real (added by the forcing)?

REPLY [9 votes]: There is a more general result proved by Groszek and Slaman which says that if there is a nonconstructible real, then every perfect set has a nonconstructible element. The constructibility can be replaced with any inner models.