A deep learning method for the particle trajectory reconstruction with the DAMPE experiment is presented. The developed algorithms constitute the first fully machine-learned track reconstruction pipeline for space astroparticle missions. Significant performance improvements over the standard hand-engineered algorithms are demonstrated. Thanks to the better accuracy, the developed algorithms facilitate the identification of the particle absolute charge with the tracker in the entire energy range, opening a door to the measurements of cosmic-ray proton and helium spectra at extreme energies, towards the PeV scale, hardly achievable with the standard track reconstruction methods. In addition, the developed approach demonstrates an unprecedented accuracy in the particle direction reconstruction with the calorimeter at high deposited energies, above several hundred GeV for hadronic showers and above a few tens of GeV for electromagnetic showers.