Here is a counterexample. Let $X$ be the ordinal $\omega^2+1$, which as a topological space is the same as infinitely many convergent sequences, whose limit points converge. This space is homeomorphic to a countable closed subset of the unit interval and is therefore completely metrizable.
Let $D$ be the isolated points of $X$, which is exactly the set of successor ordinals below $\omega^2$. This is dense, since the closure adds the missing limit ordinals. Let $f$ be the function that interleaves two successive sequences together into one. That, we combine the successor ordinals in the interval $[\omega\cdot 2n,\omega\cdot 2(n+1))$ to those in $[\omega\cdot n,\omega\cdot(n+1))$ by an injective function that simply interleaves the two sequences into one. This is injective on $D$ and also surjective. The function $f$ extends continuously to $X$ by mapping the limit points of the successive sequences, $\omega\cdot2n$ and $\omega\cdot2(n+1)$ both to $\omega\cdot n$, and $\omega^2\mapsto \omega^2$. Note that the extension $f$ is not injective.