The possibility of fabricating a polarization-maintaining Bragg fiber has been studied. It is shown that violation of the cylindrical symmetry of a Bragg mirror in most cases results in a sharp increase in optical loss, which is caused by resonance transmission through the Bragg mirror at wavelengths near the cutoffs of the modes of the high-index rings with a nonzero azimuthal index. It is shown that placing stress-applied parts or air holes inside the Bragg fiber core allows one to avoid this effect. A polarization-maintaining Bragg fiber with perfect light confinement in the core is demonstrated for the first time to our knowledge.