Let $f:\bar{B}(0,1)\rightarrow \bar{B}(0,1)$ (with $\bar{B}(0,1)=\{x\in\mathbb{R}^2| \|x\|\le1\}$) be a continuous map so that $f\restriction_{S^1}=Id\restriction_{S^1}$, that is, $f(x)=x$ $\forall x$ with $\|x\|=1$. Using a Corollary of the No Retraction Theorem, it follows that $f$ must be surjective. I assume $f$ is not necessarily injective. However, I couldn't find any counterexample to such a non-injective $f$.
In any case, for a certain physics problem, this $f$ satisfies the above conditions and, additionally, is twice continuously differentiable and maps $(0,1)\times\{0\}$ to itself bijectively. I'd need this $f$ to be indeed injective for this problem, but I could neither prove it nor come up with a counterexample. Any ideas?