How can I write a perfect set as $c$ many pairwise disjoint perfect sets

Question

I have been thinking about this question but I have not get complete answer yet. the question is Let $P$ a nonempty perfect subsets of $\mathbb R$ then $P$ can be written as continuum many pairwise disjoint perfect sets. A partial answer that I got, if we restrict perfect set $P$ to nowhere dense set, then it will be totally disconnected. So $P$ and $P\times P$ are homeomorphic. $P=\bigcup_{r\in P} (\{r\}\times P)$ as we need. My question is How about $P$ is not nowhere dense set, that is , it contains an open interval set. Any help will be useful.