  We prove an $\epsilon$-regularity theorem for vector-valued p-harmonic maps,
which are critical with respect to a partially free boundary condition, namely
that they map the boundary into a round sphere.
  This does not seem to follow from the reflection method that Scheven used for
harmonic maps with free boundary (i.e., the case $p=2$): the reflected equation
can be interpreted as a $p$-harmonic map equation into a manifold, but the
regularity theory for such equations is only known for round targets.
  Instead, we follow the spirit of the last-named author's recent work on free
boundary harmonic maps and choose a good frame directly at the free boundary.
This leads to growth estimates, which, in the critical regime $p=n$, imply
H\"older regularity of solutions. In the supercritical regime, $p < n$, we
combine the growth estimate with the geometric reflection argument: the
reflected equation is super-critical, but, under the assumption of growth
estimates, solutions are regular.
  In the case $p<n$, for stationary $p$-harmonic maps with free boundary, as a
consequence of a monotonicity formula we obtain partial regularity up to the
boundary away from a set of $(n-p)$-dimensional Hausdorff measure.
