Non-archimedean geometry is geometry over non-archimedean fields. While the concrete results are quite different, the basic formalism of algebraic schemes and formal schemes over a non-archimedean field $K$ is the special case of the standard formalism over any field. The “correct” analytic geometry over non-archimedean field, however, is not a straightforward analogue of the complex analytic case. As Tate noticed, the sheaf of $K$-valued functions which can be locally written as converging power series over the affine space $K^n$ is too big (too many analytic functions) due to the fact that the underlying topological space is totally disconnected. Also there are very few $K$-analytic manifolds. This naive approach paralleling the complex analytic geometry is called by Tate wobbly $K$-analytic varieties an, apart from the case of non-archimedean local fields it is of little use. For this reason Tate introduced a better $K$-algebra of analytic functions, locally takes its maximal spectrum and made a Grothendieck topology which takes into account just a certain smaller set of open covers; this topology is viewed as rigidified, hence the varieties based on gluing in this approach is called rigid analytic geometry. Raynaud has shown how some classes of rigid $K$-varieties can be realized as generic fibers of formal schemes over the ring of integers of $K$; this is called a formal model of a rigid variety. Different formal models are birationally equivalent, more precisely they are related via admissible blow-ups. Later more sophisticated approaches appeared:
For literature on specific approaches see the $n$Lab entries Berkovich analytic space, adic space, global analytic geometry, rigid analytic geometry, Huber space, perfectoid space.
A nice survey is in
A standard textbook on rigid analytic geometry is
For comparison see
Brian Conrad, Several approaches to non-archimedean geometry, lectures at Arizona winter school 2007, pdf
Kiran S. Kedlaya, Reified valuations and (re)adic spectra, arxiv/1309.0574
Kazuhiro Fujiwara, Fumiharu Kato, Foundations of Rigid Geometry I, arxiv/1308.4734
Oren Ben-Bassat, Kobi Kremnizer, Non-Archimedean analytic geometry as relative algebraic geometry (arXiv:1312.0338)
S. Bosch, U. Güntzer, Reinhold Remmert, Non-Archimedean Analysis – A systematic approach to rigid analytic geometry, 1984 (pdf)