  For minimizers in a geometrically nonlinear Cosserat model for micropolar
elasticity of continua, we prove interior H\"older regularity, up to isolated
singular points that may be possible if the exponent $p$ from the model is $2$
or in $(\frac{32}{15},3)$. The obstacle to full continuity turns out to be the
existence of certain minimizing homogeneous $p$-harmonic maps to $S^3$. For
those, we slightly improve existing regularity theorems in order to achieve our
result on the Cosserat model.
