  This paper is concerned with the Cauchy problem for an energy-supercritical
nonlinear wave equation in odd space dimensions that arises in equivariant
Yang-Mills theory. In each dimension, there is a self-similar finite-time
blowup solution to this equation known in closed form. It will be proved that
this profile is stable in the whole space under small perturbations of the
initial data. The blowup analysis is based on a recently developed coordinate
system called hyperboloidal similarity coordinates and depends crucially on
growth estimates for the free wave evolution, which will be constructed
systematically for odd space dimensions in the first part of this paper. This
allows to develop a nonlinear stability theory beyond the singularity.
