  Evidence for the presence of short-lived radioactive isotopes when the Solar
System formed is preserved in meteorites, providing insights into the
conditions at the birth of our Sun. A low-mass core-collapse supernova had been
postulated as a candidate for the origin of $^{10}$Be, reinforcing the idea
that a supernova triggered the formation of the Solar System. We present a
detailed study of the production of $^{10}$Be by the $\nu$ process in
supernovae, which is very sensitive to the reaction rate of the major
destruction channel, $^{10}$Be(p,$\alpha$)$^{7}$Li. With data from recent
nuclear experiments that show the presence of a resonant state in $^{11}$B at
$\approx$~193 keV, we derive new values for the $^{10}$Be(p,$\alpha$)$^{7}$Li
reaction rate which are significantly higher than previous estimates. We show
that, with the new $^{10}$Be(p,$\alpha$)$^{7}$Li reaction rate, a low mass CCSN
is unlikely to produce enough $^{10}$Be to explain the observed $^{10}$Be/
$^{9}$Be ratio in meteorites, even for a wide range of neutrino spectra
considered in our models. These findings point towards spallation reactions
induced by solar energetic particles in the early Solar System as the origin of
$^{10}$Be.
