We study the high-energy properties of GRB 181123B, a short gamma-ray burst (sGRB) at redshift $zapprox$1.75. We show that, despite its nominal short duration with $T_{90}<$2 s, this burst displays evidence of a temporally extended emission (EE) at high energies and that the same trend is observed in the majority of sGRBs at $zgtrsim$1. We discuss the impact of instrumental selection effects on the GRB classification, stressing that the measured $T_{90}$ is not an unambiguous indicator of the burst physical origin. By examining their environment (e.g. stellar mass, star formation, offset distribution), we find that these high-$z$ sGRBs share many properties of long GRBs at a similar distance and are consistent with a short-lived progenitor system. If produced by compact binary mergers, these sGRBs with EE may be easier to localize at large distances and herald a larger population of sGRBs in the early universe.