  We study $p$--harmonic maps with Dirichlet boundary conditions from a planar
domain into a general compact Riemannian manifold. We show that as $p$
approaches $2$ from below, they converge up to a subsequence to a minimizing
singular renormalizable harmonic map. The singularities are imposed by
topological obstructions to the existence of harmonic mappings; the location of
the singularities being governed by a renormalized energy. Our analysis is
based on lower bounds on growing balls and also yields some uniform weak-$L^p$
bounds (also known as Marcinkiewicz or Lorentz $L^{p,\infty}$).
