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Long-duration gamma-ray burst (GRB) afterglow observations offer cutting-edge opportunities to characterise the star formation history of the Universe back to the epoch of reionisation, and to measure the chemical composition of interstellar and intergalactic gas through absorption spectroscopy. The main barrier to progress is the low efficiency in rapidly and confidently identifying which bursts are high redshift ($z > 5$) candidates before they fade, as this requires low-latency follow-up observations at near-infrared wavelengths (or longer) to determine a reliable photometric redshift estimate. So far this task has been performed by instruments on the ground, but sky visibility and weather constraints limit the number of GRB targets that can be observed and the speed at which follow-up is possible. In this work we develop a Monte Carlo simulation framework to investigate an alternative approach based on the use of a rapid-response near-infrared nano-satellite, capable of simultaneous imaging in four bands from $0.8$ to $1.7μ$m (a mission concept called SkyHopper). We find that such a nano-satellite is capable of detecting in the H band (1.6 $μ$m) $72.5\% \pm 3.1\%$ of GRBs concurrently observable with the Swift satellite via its UVOT instrument (and $44.1\% \pm 12.3\%$ of high redshift ($z>5$) GRBs) within 60 minutes of the GRB prompt emission. This corresponds to detecting $\sim 55$ GRB afterglows per year, of which 1-3 have $z > 5$. These rates represent a substantial contribution to the field of high-$z$ GRB science, as only 23 $z > 5$ GRBs have been collectively discovered by the entire astronomical community over the last $\sim 24$ years. Additionally, we find that launching a mini-constellation of 3 near-infrared nano-satellites would increase the detection fraction of afterglows to $\sim 83\%$ and substantially reduce the latency in the photometric redshift determination.