Electronic transport in ferromagnetic ballistic conductors is predicted to exhibit ballistic anisotropic magnetoresistance-a change in the ballistic conductance with the direction of magnetization. This phenomenon originates from the effect of the spin-orbit interaction on the electronic band structure which leads to a change in the number of bands crossing the Fermi energy when the magnetization direction changes. We illustrate the significance of this phenomenon by performing ab initio calculations of the ballistic conductance in ferromagnetic Ni and Fe nanowires which display a sizable ballistic anisotropic magnetoresistance when magnetization changes direction from parallel to perpendicular to the wire axis.