If $k$ is a field that is complete with respect to some ultrametric valuation, then there is the "GAGR" (i.e. géométrie algébrique et géométrie rigide) theorem. A succinct explanation (in French, without proof) is:
Jarraud, Pierre. À propos de G. A. G. R.. Groupe de travail d'analyse ultramétrique 11 (1983-1984): 1-4.See [link][1].
The original source for the result, listed as [4] in Jarraud's note, seems to be:
Ursula Köpf, Über eigentliche Familien algebraischer Varietäten über affinoïden Räumen, Schriftenreihe Univ. Münster, 2. Serie, Heft 7 (1974).