  Gamma-ray observations have long been used to constrain the properties of
dark matter (DM), with a strong focus on weakly interacting massive particles
annihilating through velocity-independent processes. However, in the absence of
clear-cut observational evidence for the simplest candidates, the interest of
the community in more complex DM scenarios involving a velocity-dependent
cross-section has been growing steadily over the past few years. We present the
first systematic study of velocity-dependent DM annihilation (in particular
$p$-wave annihilation and Sommerfeld enhancement) in a variety of astrophysical
objects, not only including the well-studied Milky Way dwarf satellite
galaxies, but nearby dwarf irregular galaxies and local galaxy clusters as
well. Particular attention is given to the interplay between velocity
dependence and DM halo substructure. Uncertainties related to halo mass,
phase-space and substructure modelling are also discussed in this
velocity-dependent context. We show that, for $s$-wave annihilation, extremely
large subhalo boost factors are to be expected, up to $10^{11}$ in clusters and
up to $10^6-10^7$ in dwarf galaxies where subhalos are usually assumed not to
play an important role. Boost factors for $p$-wave annihilation are smaller but
can still reach $10^3$ in clusters. The angular extension of the DM signal is
also significantly impacted, with e.g. the cluster typical emission radius
increasing by a factor of order 10 in the $s$-wave case. We also compute the
signal contrast of the objects in our sample with respect to annihilation
happening in the Milky Way halo. Overall, we find that the hierarchy between
the brightest considered targets depends on the specific details of the assumed
particle-physics model.
