Suppose that

- $E$ and $F$ are complex Banach spaces and $U\subset E$ and $V\subset F$ are open subses.
- $f\colon U\to V$ is analytic
- $f\colon U\to V$ is bijective

Is $f$ bi-analytic? (i.e. is its inverse $f^{-1}\colon V\to U$ analytic?)

I heard this result is known but I could not find a reference

EDIT: As pointed out by @abx my original question in the real analytic setting has trivial counter examples.