The category of pro-étale vector bundles on a proper rigid-analytic variety X over C is equivalent to the category of Higgs bundles on the eh-site of X.
flattening techniques, Mathematische Annalen 296 (1993), 403--429
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A $p$-adic Simpson correspondence for singular rigid-analytic varieties
The category of pro-étale vector bundles on a proper rigid-analytic variety X over C is equivalent to the category of Higgs bundles on the eh-site of X.