  We introduce a variant of homotopy K-theory for Tate rings, which we call
analytic K-theory. It is homotopy invariant with respect to the analytic affine
line viewed as an ind-object of closed disks of increasing radii. Under a
certain regularity assumption we prove an analytic analog of the Bass
fundamental theorem and we compare analytic K-theory with continuous K-theory,
which is defined in terms models. Along the way we also prove some results
about the algebraic K-theory of Tate rings.
