A very nice question!

The answer is no, there cannot be a perfect set in $V[c]$
consisting entirely of ground-model reals.

Suppose towards contradiction that there is a such a set. So this
set consists of the paths through a certain perfect tree $T\subset
2^{<\omega}$ in $V[c]$.

Note, as an easy first case, that we cannot have $T$ in the ground
model $V$, for in this case we may consider the branch through $T$
specified by going left-or-right at the splitting nodes of $T$
according to the digits of $c$. From this branch and $T$ we could
reconstruct $c$ in the ground model, which is a contradiction.

For the general case, fix a name $\dot T$ for the tree, and suppose
that the trivial condition forces $\dot T$ is a perfect tree all of
whose branches are in the ground model. For any particular branch
$z$ through $T$, there is a finite initial segment of $c$ that
forces that $\check z$ is a branch through $\dot T$. Since there are
uncountably many branches through $T$ in $V[c]$, there must be some
fixed condition $p=c\upharpoonright n$ forcing uncountably many
reals $z$ as branches through $\dot T$.

Let $S$ be the tree of finite binary sequences that $p$ forces in
$\dot T$. So $S$ is in $V$, and $S$ has uncountably many branches
in $V$. So by the Cantor-Bendixson theorem, it follows that $S$ has a perfect subtree $S_0$ in $V$, since every uncountable closed set has a perfect subset. But in this case, we have a perfect tree $S_0$ in $V$ all of whose branches in
$V[c]$ are in $V$. And that was the first case we ruled out.

So there can be no such set. QED

Alternative summary of the argument. If $C$ is a perfect set of ground-model reals, let $\dot C$ be a name for it. By pigeon-hole principle, there is a single condition $p$ forcing $\check z\in\dot C$ for uncountably many reals $z$. let $C_p$ be the set of $z$ for which $p\Vdash\check z\in\dot P$. So this is an uncountable closed set in the ground model, which therefore contains a perfect set, the branches through a splitting tree in $V$. But any such tree has branches in $V[c]$ that are not in $V$.