  Dwarf spheroidal galaxies provide well-known challenges to the standard cold
and collisionless dark matter scenario: The too-big-to-fail problem, namely the
mismatch between the observed mass enclosed within the half-light radius of
dwarf spheroidals and cold dark matter N-body predictions; The hints for inner
constant-density cores. While these controversies may be alleviated by baryonic
physics and environmental effects, revisiting the standard lore of cold and
collisionless dark matter remains an intriguing possibility. Self-interacting
dark matter may be the successful proposal to such a small-scale crisis.
Self-interactions correlate dark matter and baryon distributions, allowing for
constant-density cores in low surface brightness galaxies. Here we report the
first data-driven study of the too-big-to-fail of Milky Way dwarf spheroidals
within the self-interacting dark matter paradigm. We find good description of
stellar kinematics and compatibility with the concentration-mass relation from
recent pure cold dark matter simulations. Within this concentration-mass
relation, a subset of Milky Way dwarfs are well fitted by cross sections of
0.5-3 cm$^2$/g, while others point to values greater than 10 cm$^2$/g.
