  We define a new category of non-archimedean analytic spaces over a complete
discretely valued field, which we call uniformly rigid. It extends the category
of rigid spaces, and it can be described in terms of bounded functions on
products of open and closed polydiscs. We relate uniformly rigid spaces to
their associated classical rigid spaces, and we transfer various constructions
and results from rigid geometry to the uniformly rigid setting. In particular,
we prove an analog of Kiehl's patching theorem for coherent ideals, and we
define the uniformly rigid generic fiber of a formal scheme of formally finite
type. This uniformly rigid generic fiber is more intimately linked to its model
than the classical rigid generic fiber obtained via Berthelot's construction.
