  We present the discovery of the optical afterglow and host galaxy of the {\it
Swift} short-duration gamma-ray burst, GRB\,181123B. Observations with
Gemini-North starting at $\approx 9.1$~hr after the burst reveal a faint
optical afterglow with $i\approx25.1$~mag, at an angular offset of 0.59 $\pm$
0.16$''$ from its host galaxy. Using $grizYJHK$ observations, we measure a
photometric redshift of the host galaxy of $z = 1.77^{+0.30}_{-0.17}$. From a
combination of Gemini and Keck spectroscopy of the host galaxy spanning
4500-18000~\AA , we detect a single emission line at 13390~\AA, inferred as
H$\beta$ at $z = 1.754 \pm 0.001$ and corroborating the photometric redshift.
The host galaxy properties of GRB\,181123B are typical to those of other SGRB
hosts, with an inferred stellar mass of $\approx 1.7 \times
10^{10}\,M_{\odot}$, mass-weighted age of $\approx 0.9$~Gyr and optical
luminosity of $\approx 0.9L^{*}$. At $z=1.754$, GRB\,181123B is the most
distant secure SGRB with an optical afterglow detection, and one of only three
at $z>1.5$. Motivated by a growing number of high-$z$ SGRBs, we explore the
effects of a missing $z>1.5$ SGRB population among the current {\it Swift}
sample on delay time distribution models. We find that log-normal models with
mean delay times of $\approx 4-6$~Gyr are consistent with the observed
distribution, but can be ruled out to $95\%$ confidence with an additional
$\approx1-5$~{\it Swift} SGRBs recovered at $z>1.5$. In contrast, power-law
models with $\propto$ $t^{-1}$ are consistent with the redshift distribution
and can accommodate up to $\approx30$ SGRBs at these redshifts. Under this
model, we predict that $\approx 1/3$ of the current {\it Swift} population of
SGRBs is at $z>1$. The future discovery or recovery of existing high-$z$ SGRBs
will provide significant discriminating power on their delay time
distributions, and thus their formation channels.
