Identity theorem in $p$-adic geometry/analysis

Question

If one wants to do $p$-adic analysis and geometry, it is often bad so adapt "naively" complex analytic ideas, basically because $mathbb{Q}_p$ is disconnected. The modern approach to this is, to my knowledge, the theory of rigid analytic spaces and Berkovich spaces. For instance in the theory of Berkovich spaces, the maximum modulus principle holds. I'm curious what we can say about the Identity theorem. Does it hold in any of the $p$-adic geometries?
I've been told to think of rigid analytic spaces as varieties and (Huber's-)adic spaces as schemes with Berkovich being closer to an "analytic" object, so probably I'd expect it (if at all) to hold there.

Jérôme Poineau · Answer

Let me assume that you have an analytic function $f$ defined on a closed one-dimensional unit disc, which is to say the spectrum of the Tate algebra $k{T}$. Then, Weierstrass preparation theorem tells you that $f$ may be written as a product of a polynomial $P$ and a nowhere vanishing function. If $P$ is non-zero, it has only finitely many zeroes, and so does $f$.
As you see, it can really be turned into a statement about Tate algebras, so the precise theory you want to work with does not really matter.
Please, let me know if you had a more general context in mind.

Identity theorem in $p$-adic geometry/analysis

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