  We define a notion of global analytic space with overconvergent structure
sheaf. This gives an analog on a general base Banach ring of Grosse-Kloenne's
overconvergent p-adic spaces and of Bambozzi's generalized affinoid varieties
over R. This also gives an affinoid version of Berkovich's and Poineau's global
analytic spaces. This affinoid approach allows the introduction of a notion of
strict global analytic space, that has some relations with the ideas of
Arakelov geometry, since the base extension along the identity morphism on Z
(from the archimedean norm to the trivial norm) sends a strict global analytic
space to a usual scheme over Z, that we interpret here as a strict analytic
space over Z equipped with its trivial norm. One may also interpret some
particular analytification functors as mere base extensions. We use our
categories to define overconvergent motives and an overconvergent stable
homotopy theory of global analytic spaces. These have natural Betti, de Rham
and pro-\'etale realizations. Motivated by problems in global Hodge theory and
integrality questions in the theory of special values of arithmetic
L-functions, we also define derived overconvergent global analytic spaces and
their (derived) de Rham cohomology. Finally, we use Toen and Vezzosi's derived
geometric methods to define a natural (integral) Chern character on analytic
Waldhausen's K-theory with values in analytic cyclic homology. The
compatibility of our constructions with Banach base extensions gives new
perspectives both on global analytic spaces and on the various realizations of
the corresponding motives.
