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Let $\Omega$ be a non-archimedean complete field, $n\in\mathbb N$ and $f:\Omega^n\to\Omega^n$ be an injective analytic map. Is the application $f$ open?

In the complex case, this is a consequence of a Remmert theorem, but in the non-archimedean case, I do not know.

Thanks in advance for any answer.

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Let $p\equiv 2\pmod{3}$ so that $\mathbb{Q}_p$ contains no cube root of unity other than $1$. The map $x \mapsto x^3$ from $\mathbb{Q}_p$ to itself is injective, yet its image contains no neighborhood of $0$ (because only points whose valuation is multiple of $3$).

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