higher geometry / derived geometry
geometric little (∞,1)-toposes
geometric big (∞,1)-toposes
function algebras on ∞-stacks?
derived smooth geometry
The special case of non-archimedean analytic geometry based on p-adic numbers.
Survey is in
Bernard Le Stum, One century of $p$-adic geometry – From Hensel to Berkovich and beyond, talk notes, June 2012 (pdf)
Peter Scholze, $p$-adic geometry, ICM report (arXiv:1712.03708)
Peter Scholze, Jared Weinstein, Berkeley lectures on $p$-adic geometry, pdf.