  A tentative excess in the electron spectrum at 1.4 TeV was recently reported
by the DArk Matter Particle Explorer (DAMPE). A non-astrophysical scenario in
which dark matter particles annihilate or decay in a local clump has been
invoked to explain the excess. If $e^\pm$ annihilation channels in the final
states are mediated by left-handed leptons as a component in the $SU(2)_L$
doublet, neutrinos with similar energies should have been simultaneously
produced. We demonstrate that generic dark matter models can be decisively
tested by the existing IceCube data. In case of a non-detection, such models
would be excluded at the $5\sigma$ level by the five-year data for a point-like
source and by the ten-year data for an extended source of dark matter particles
with left-handed leptons.
