Some notes by Kedlaya. Some motivation.
http://nlab.mathforge.org/nlab/show/rigid+analytic+geometry
I think rigid analytic geometry is mentioned as an example in Higher topos theory
Ask Michael for other notes.
What is rigid analytic geometry? One ref is Bosch, Guntzer and Remmert: Non-archimedean analysis. Springer, 1984.
MR review of the book of Huber http://www.ams.org/mathscinet-getitem?mr=1734903. At least five frameworks for non-Archimedean anlytic geometry: Tate’s rigid analytic approach, Raynaud’s formal scheme approach, Berkovich spaces, and Fujiwara’s Zariski-Riemann spaces. To this one should maybe add perfectoid spaces, or maybe they are a subclass of adic spaces. For the relation between these notions, see the review and Scholze’s first IHES lecture.
nLab page on Rigid analytic geometry