Does there exist a Borel (or even continuous) function $f:\mathcal{C}\to\mathcal{C}$, where $\mathcal{C}$ is the Cantor set (or Cantor space $2^\omega$) such that for every nonempty closed perfect set $P\subseteq\mathcal{C}$, $f|P$ maps surjectively onto $\mathcal{C}$?

Such functions (on $\mathbb{R}$) are called *perfectly everywhere surjective* here: https://core.ac.uk/download/pdf/83599431.pdf, but the maps constructed there rely on, in essence, a well-ordering of $\mathbb{R}$ and are likely far from any kind of measurability.

Hoping that the self-similarity of the Cantor set could be exploited here.