  The signals observed at the direct detection experiments DAMA, CoGeNT and
CRESST could be explained by light WIMPs with sizeable spin-independent cross
sections with nucleons. The capture and subsequent annihilation of such
particles in the sun would induce neutrino signals in the GeV range which may
be observed at Super-Kamiokande. We determine the rate of upward stopping muons
and fully contained events at Super-Kamiokande for various possible WIMP
annihilation channels. This allows us to provide strong constraints on the
cross section of WIMPs with nucleons. We find that the DAMA and CoGeNT signals
are inconsistent with standard thermal WIMPs annihilating dominantly into
neutrino or tau pairs. We also provide limits for spin-dependent WIMP nucleus
scattering for masses up to 80 GeV. These exclude the DAMA favored region if
WIMPs annihilate even subdominantly into neutrinos, taus, bottoms or charms.
