The properties of the antikaons in nuclear matter are investigated from a chiral unitary approach which incorporates the s- and p-waves of the K¯N interaction. To obtain the in-medium meson-baryon amplitudes we include, in a self-consistent way, Pauli blocking effects, meson self-energies corrected by nuclear short-range correlations and baryon binding potentials. We pay special attention to investigating the validity of the on-shell factorization, showing that it cannot be applied in the evaluation of the in-medium corrections to the p-wave amplitudes. In nuclear matter at saturation energy, the Λ and Σ develop an attractive potential of about -30 MeV, while the Σ∗ pole remains at the free space value although its width gets sensibly increased to about 80 MeV. The antikaon also develops a moderate attraction that does not support the existence of very deep and narrow bound states, confirming the findings of previous self-consistent calculations.