Abstract We present Chandra ACIS-S imaging spectroscopy results of the extended (1.″5–8″, 300–1600 pc) hard X-ray emission of NGC 5728, the host galaxy of a Compton-thick active galactic nucleus. We find spectrally and spatially resolved features in the Fe K α complex (5.0–7.5 keV) redward and blueward of the neutral Fe line at 6.4 keV in the extended narrow-line region bicone. A simple phenomenological fit of a power law plus Gaussians gives a significance of 5.4 σ and 3.7 σ for the red and blue wings, respectively. Fits to a suite of physically consistent models confirm a significance of ≥3 σ for the red wing. The significance of the blue wing may be diminished by the presence of rest-frame highly ionized Fe xxv and Fe xxvi lines (1.4 σ –3.7 σ range). A detailed investigation of the Chandra ACIS-S point-spread function and comparison with the observed morphology demonstrates that these red and blue wings are radially extended (∼5″, ∼1 kpc) along the optical bicone axis. If the wing emission is due solely to redshifted and blueshifted high-velocity neutral Fe K α , then the implied line-of-sight velocities are +/− ∼0.1 c , and their fluxes are consistent with being equal. A symmetric high-velocity outflow is then a viable explanation. This outflow has deprojected velocities ∼100 times larger than the outflows detected in optical spectroscopic studies, potentially dominating the kinetic feedback power.