  We study the negative gradient flow of the spinorial energy functional
(introduced by Ammann, Wei{\ss}, and Witt) on 3-dimensional Berger spheres. For
a certain class of spinors we show that the Berger spheres collapse to a
2-dimensional sphere. Moreover, for special cases, we prove that the
volume-normalized standard 3-sphere together with a Killing spinor is a stable
critical point of the volume-normalized version of the flow. Our results also
include an example of a critical point of the volume-normalized flow on the
3-sphere, which is not a Killing spinor.
