We present the discovery of the optical afterglow and host galaxy of the Swift short-duration gamma-ray burst (SGRB) GRB 181123B. Observations with Gemini-North starting 9.1 hr after the burst reveal a faint optical afterglow with i 25.1 mag at an angular offset of 0 59 0 16 from its host galaxy. Using grizYJHK observations, we measure a photometric redshift of the host galaxy of . From a combination of Gemini and Keck spectroscopy of the host galaxy spanning 4500-18000 , we detect a single emission line at 13390 , inferred as Hβ at z = 1.754 0.001 and corroborating the photometric redshift. The host galaxy properties of GRB 181123B are typical of those of other SGRB hosts, with an inferred stellar mass of 9.1 × 109 M , a mass-weighted age of 0.9 Gyr, and an optical luminosity of 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 Swift sample on delay time distribution (DTD) models. We find that lognormal models with mean delay times of 4-6 Gyr are consistent with the observed distribution but can be ruled out to 95% confidence, with an additional one to five Swift SGRBs recovered at z > 1.5. In contrast, power-law models with ∝t−1 are consistent with the redshift distribution and can accommodate up to 30 SGRBs at these redshifts. Under this model, we predict that 1/3 of the current 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 DTDs and thus their formation channels.