Exsistence of branch of a complex valued function

Question

Does $ \exists $ $f\in Hol (\mathbb{C}\setminus${0})$)$ such that $f(z)^2$ = $ z $ $ \forall$ $z\in \mathbb{C}\setminus${0}.

Answer 1

Any such function would have a continuous and therefore holomorphic extension to $0$ by setting $f(0) = 0.$ (since $|f(z)|^2$ becomes arbitrarily small as $z \rightarrow 0$)

This extension $f$ maps the open unit disc $\mathbb{E}$ to itself and has $f(0) = 0$, but it violates Schwarz's lemma since $|f(z)| \le |z|$ is not true.