The very small accretion disks in ultra-compact X-ray binaries (UCXBs) are special laboratories in which to study disk accretion and outflows. We report on three sets of new (250 ks total) and archival (50 ks) Chandra/HETG observations of the dipping neutron-star X-ray binary 4U 1916$-$053, which has an orbital period of $Psimeq 50$~minutes. We find that the bulk of the absorption in all three spectra originates in a disk atmosphere that is redshifted by $vsimeq 220-290$ $text{km}$ $text{s}^{-1}$, corresponding to the gravitational redshift at radius of $R sim 1200$ $GM/{c}^{2}$. This shift is present in the strongest, most highly ionized lines (Si XIV and Fe XXVI), with a significance of 5$sigma$. Absorption lines observed during dipping events (typically associated with the outermost disk) instead display no velocity shifts and serve as a local standard of rest, suggesting that the redshift is intrinsic to an inner disk atmosphere and not due to radial motion in the galaxy or a kick. In two spectra, there is also evidence of a more strongly redshifted component that would correspond to a disk atmosphere at $R sim 70$ $GM/{c}^{2}$; this component is significant at the 3$sigma$ level. Finally, in one spectrum, we find evidence of disk wind with a blue shift of $v = {-1700}^{+1700}_{-1200}$ $text{km}$ $text{s}^{-1}$. If real, this wind would require magnetic driving.
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