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:
<blockquote>
Jarraud, Pierre. <a href="http://eudml.org/doc/91922"><i>À propos de G.A.G.R..</i></a> Groupe de travail d'analyse ultramétrique <b>11</b> (1983-1984): 1-4.
</blockquote>

The original source for the result, listed as $[4]$ in Jarraud's note, seems to be:
<blockquote>
Köpf, Ursula. <i>Über eigentliche Familien algebraischer Varietäten über affinoïden Räumen</i>, Schriftenreihe Univ. Münster, 2. Serie, Heft 7 (1974).
</blockquote>


  [1]: http://eudml.org/doc/91922