  Sommerfeld-enhanced dark matter (DM) annihilation through $s$-wave has been
widely considered as a consistent explanation for both the observed cosmic-ray
(CR) positron excess and the DM thermal relic density. However, as the $s$-wave
Sommerfeld-enhanced annihilation cross section increases monotonically with
decreasing DM velocity, severe constraints appear from the data of gamma rays
from dwarf spheroidal satellite galaxies (dSphs) and the cosmic microwave
background (CMB), as the relevant typical DM velocities are even lower than
that in the Galactic halo. In this work, we consider Sommerfeld-enhanced
$p$-wave DM annihilation where the DM annihilation cross section can be
enhanced at certain velocities but eventually be highly suppressed when the DM
velocities become extremely low. We calculate the velocity-dependent
astrophysics factors ($J$-factors) for the Sommerfeld-enhanced $p$-wave DM
annihilation for fifteen nearby dSphs. Taking the channel of DM annihilating
into $4\mu$ through two light mediators as an example, we show that there are
parameter regions where this mechanism can account for the CR positron excess
and the DM relic density, while being compatible with the constraints from
dSphs gamma rays measured by Fermi-LAT and that from the CMB measured by
PLANCK.
