  Recent measurements of very low-energy ($p_{L}<100$ MeV/c) $\bar p$
annihilation on light nuclei reveal apparent suppression of annihilation upon
increasing the atomic charge $Z$ and mass number $A$. Using $\bar p$-nucleus
optical potentials $V_{{\rm opt}}$, fitted to $\bar p$-atom energy-shifts and
-widths, we resolve this suppression as due to the strong effective repulsion
produced by the very absorptive $V_{{\rm opt}}$. The low-energy $\bar
p$-nucleus wavefunction is kept substantially outside the nuclear surface and
the resulting reaction cross section saturates as function of the strength of
Im $V_{{\rm opt}}$. This feature, for $E >0$, parallels the recent prediction,
for $E < 0$, that the level widths of $\bar p$ atoms saturate and, hence, that
$\bar p$ deeply bound atomic states are relatively narrow. Antiproton
annihilation cross sections are calculated at $p_{L}=57$ MeV/c across the
periodic table, and their dependence on $Z$ and $A$ is classified and discussed
with respect to the Coulomb focussing effect at very low energies.
