Finite union of affinoid is affinoid in proper smooth rigid curves (unless it is everything)

In several papers I have found the surprising statement that finite unions of affinoid subspaces of a proper smooth and connected rigid curve are either the whole curve or again affinoid.

Could you give me a reference for this fact or help me to sketch a proof?

Thank you in advance.

• You probably want to assume the curve is connected, otherwise you may end up with the disjoint union of an affinoid curve and a projective curve. – Jérôme Poineau Feb 25 '16 at 19:59