  The Planck collaboration has recently published precise and resolved
measurements of the Sunyaev-Zel'dovich effect in Abell 1656 (the Coma cluster
of galaxies), so directly gauging the electron pressure profile in the
intracluster plasma. On the other hand, such a quantity may be also derived
from combining the density and temperature provided by X-ray observations of
the thermal bremsstrahlung radiation emitted by the plasma. We find a
model-independent tension between the SZ and the X-ray pressure, with the SZ
one being definitely lower by 15-20%. We propose that such a challenging
tension can be resolved in terms of an additional, non-thermal support to the
gravitational equilibrium of the intracluster plasma. This can be
straightforwardly included in our Supermodel, so as to fit in detail the Planck
SZ profile while being consistent with the X-ray observables. Possible origins
of the nonthermal component include cosmic-ray protons, ongoing turbulence, and
relativistic electrons; given the existing observational constraints on the
first two options, here we focus on the third. For this to be effective, we
find that the electron population must include not only an energetic tail
accelerated to gamma> 10^3 responsible for the Coma radiohalo, but also many
more, lower energy electrons. The electron acceleration is to be started by
merging events similar to those which provided the very high central entropy of
the thermal intracluster plasma in Coma.
