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Context.Grids of stellar evolution models with rotation using the Geneva stellar evolution code (GENEC) have been published for a wide range of metallicities. Aims.We introduce the last remaining grid of GENECmodels, with a metallicity ofZ = 10⁻⁵. We study the impact of this extremely metal-poor initial composition on various aspects of stellar evolution, and compare it to the results from previous grids at other metallicities. We provide electronic tables that can be used to interpolate between stellar evolution tracks and for population synthesis. Methods.Using the same physics as in the previous papers of this series, we computed a grid of stellar evolution models with GENECspanning masses between 1.7 and 500M_⊙, with and without rotation, at a metallicity ofZ = 10⁻⁵. Results.Due to the extremely low metallicity of the models, mass-loss processes are negligible for all except the most massive stars. For most properties (such as evolutionary tracks in the Hertzsprung-Russell diagram, lifetimes, and final fates), the present models fit neatly between those previously computed at surrounding metallicities. However, specific to this metallicity is the very large production of primary nitrogen in moderately rotating stars, which is linked to the interplay between the hydrogen- and helium-burning regions. Conclusions.The stars in the present grid are interesting candidates as sources of nitrogen-enrichment in the early Universe. Indeed, they may have formed very early on from material previously enriched by the massive short-lived Population III stars, and as such constitute a very important piece in the puzzle that is the history of the Universe.