We report the first detections of associated H{sc i} 21,cm absorption in Gigahertz-peaked-spectrum (GPS) sources at high redshifts, $z > 1$, using the Giant Metrewave Radio Telescope (GMRT). Our GMRT search for associated H{sc i} 21,cm absorption in a sample of 12 GPS sources yielded two new detections of absorption, towards TXS~1200+045 at $z = 1.226$ and TXS~1245$-$197 at $z = 1.275$, and five non-detections. These are only the sixth and seventh detections of associated H{sc i} 21,cm absorption in active galactic nuclei (AGNs) at $z > 1$. Both H{sc i} 21,cm absorption profiles are wide, with velocity spans between nulls of $approx 600$~km~s$^{-1}$ (TXS~1200+045) and $approx 1100$~km~s$^{-1}$ (TXS~1245$-$197). In both absorbers, the large velocity spread of the absorption and its blueshift from the AGN, suggests that it arises in outflowing neutral gas, perhaps driven by the radio jets to high velocities. We derive mass outflow rates of ${dot M} approx 32 ; {rm M}_odot$~yr$^{-1}$ (TXS~1200+045) and ${dot M} approx 18 ; {rm M}_odot$~yr$^{-1}$ (TXS~1245$-$197), comparable to the mass outflow rates seen earlier in low-redshift active galactic nuclei.
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