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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 z = 1.77+0.30-0.17. From a combination of Gemini and Keck spectroscopy of the host galaxy spanning 4500–18000 Å, we detect a single emission line at 13390 Å, 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 × 109Msun, 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.