  Bipolaronic superconductivity is an exotic pairing mechanism proposed for
materials like Ba$_{1-x}$K$_x$BiO$_3$ (BKBO); however, conclusive experimental
evidence for a (bi)polaron metallic state in this material remains elusive.
Here, we combine resonant inelastic x-ray and neutron total scattering
techniques with advanced modelling to study the local lattice distortions,
electronic structure, and electron-phonon coupling ($e$-ph) in BKBO as a
function of doping. Data for the parent compound ($x = 0$) indicates that the
electronic gap opens in predominantly oxygen-derived states strongly coupled to
a long-range ordered breathing distortion of the oxygen sublattice. Upon
doping, short-range breathing distortions and sizable ($e$-ph) coupling persist
into the superconducting regime ($x = 0.4$). Comparisons with exact
diagonalization and determinant quantum Monte Carlo calculations further
support this conclusion. Our results provide compelling evidence that BKBO's
metallic phase hosts a liquid of small (bi)polarons derived from local
breathing distortions of the lattice, with implications for understanding the
low-temperature superconducting instability
