I think that the original reference is "Zariski's Main Theorem für affinoide Kurven" by K.-H. Fieseler (Mat. Ann. 251, 1980). He proves that a finite union of affinoid domains of a one-dimensional affinoid space is affinoid, but this is probably not enough to answer your question.

More generally, though, J. Fresnel and M. Matignon prove that a quasi-compact irreducible one-dimensional rigid space is either affinoid or projective ("Sur les espaces analytiques quasi-compacts de dimension 1 sur un corps valué complet ultramétrique", Annali di Matematica Pura ed Applicata 145, 1986). You can also find a proof in chapter 6 of A. Ducros's book (see https://webusers.imj-prg.fr/~antoine.ducros/livre.html) in the language of Berkovich spaces.