The solar wind is permeated by various kinds of fluctuations ranging broadly in scales from those of the solar corona and inner heliosphere down to the local ion and electron plasma kinetic scales. The question of what rules the dissipation of magnetohydrodynamic (MHD) turbulence in the solar wind has not conclusively been answered, but remains a key research topic of space plasma physics. Here we propose a new dissipation mechanism, the proton Landau damping of the quasi-perpendicular kinetic slow mode. This mode is linked to the oblique MHD slow mode, yet has shorter wavelengths going down to the proton inertial length. The kinetic slow mode can be separated from the kinetic Alfvén mode by the Alfvén resonance parameter, the proton Landau resonance parameter, the magnetic compressibility, and the electric field polarization. Numerical simulations and in situ observations indicate that the MHD turbulent cascade preferably transfers energy in the direction perpendicular to the background magnetic field. If the kinetic slow mode is also generated and replenished by the energy cascade, this mode can lead to both perpendicular and parallel heating of the protons.