  The s- and p-wave contributions to the $\bar K N$ interaction in dense
nuclear matter are obtained using a chiral unitary approach. We perform a
self-consistent calculation of the $\bar K$ self-energy including Pauli
blocking effects, meson self-energies modified by short-range correlations and
baryon binding potentials. We find that the on-shell factorization cannot be
applied to evaluate the in-medium corrections to p-wave amplitudes.
Furthermore, the $\Lambda$ and $\Sigma$ develop a mass shift of -30 MeV at
saturation density while the $\Sigma^*$ width increases to 80 MeV. We conclude
that no deep and narrow $\bar K$ bound states could be observed.
