What would be a non-injective holomorphic function on $B(0,\epsilon)$?

Question

Let $\epsilon > 0$.

What would be a non-injective holomorphic function on $B(0,\epsilon)$ such that $f'(0)\neq 0$?

Since $f'(0)\neq 0$, there exists a neighborhood of $0$ such that $f$ is injective on that neighborhood. But it need not fill out all $B(0,\epsilon)$. So there must be an example.

Answer 1

For example $f(z) = \exp\Big(\dfrac{4\pi iz}{\varepsilon}\Big)$. Then $f(\varepsilon/2) = f(0) = 1$.