  We study the exact spherically symmetric solutions in a class of
Lorentz-breaking massive gravity theories, using the effective-theory approach
where the graviton mass is generated by the interaction with a suitable set of
Stuckelberg fields. We find explicitly the exact black hole solutions which
generalizes the familiar Schwarzschild one, which shows a non-analytic hair in
the form of a power-like term r^\gamma. For realistic self-gravitating bodies,
we find interesting features, linked to the effective violation of the Gauss
law: i) the total gravitational mass appearing in the standard 1/r term gets a
multiplicative renormalization proportional to the area of the body itself; ii)
the magnitude of the power-like hairy correction is also linked to size of the
body. The novel features can be ascribed to presence of the goldstones fluid
turned on by matter inside the body; its equation of state approaching that of
dark energy near the center. The goldstones fluid also changes the matter
equilibrium pressure, leading to an upper limit for the graviton mass, m <~
10^-28 - 10^29 eV, derived from the largest stable gravitational bound states
in the Universe.
