I think that recursion theory gives a clearer way to answer the question. If $f$ is a Borel function, then it is a hyperarithmetic reduction relative to a real, say $x$. Then fix any "regular" forcing (random forcing, Cohen forcing etc) which always produce ``powerless" generic reals, we may pick up a perfect set $P$ of such generic reals. Now restricted to $P$, $f$ cannot range over the whole Cantor space. For example, $\mathscr{O}^x$, the hyperjump of $x$, does not belong the range.

The method can be push up to more set theoretical. For example, by almost the same argument, it can be shown that in Solovay model, there is such function.

I think that recursion theory gives a clearer way to answer the question. If $f$ is a Borel function, then it is a hyperarithmetic reduction relative to a real, say $x$. Then fix any "regular" forcing (random forcing, Cohen forcing etc) which always produce ``powerless" generic reals, we may pick up a perfect set $P$ of such generic reals. Now restricted to $P$, $f$ cannot range over the whole Cantor space. For example, $\mathscr{O}^x$, the hyperjump of $x$, does not belong the range.

The method can be push up to more set theoretical. For example, by almost the same argument, it can be shown that in Solovay model, there is no such function.