  The present {\em ab initio} study shows that in BaBiO$_3$, Bi$^{3+}$ sites
can trap two holes from the valence band to form Bi$^{5+}$ cations. The
trapping is accompanied by large local lattice distortions, therefore the
composite particle consisting of the electronic-hole and the local lattice
phonon field forms a polaron. Our study clearly shows that even $sp$ elements
can trap carriers at lattice sites, if local lattice relaxations are
sufficiently large to screen the localised hole. The derived model describes
all relevant experimental results, and settles the issue of why hole doped
BaBiO$_3$ remains semiconducting upon moderate hole doping.
