Large-area lithium-drifted silicon (Si(Li)) detectors, operable 150{deg}C above liquid nitrogen temperature, have been developed for the General Antiparticle Spectrometer (GAPS) balloon mission and will form the first such system to operate in space. These 10 cm-diameter, 2.5 mm-thick multi-strip detectors have been verified in the lab to provide <4 keV FWHM energy resolution for X-rays as well as tracking capability for charged particles, while operating in conditions (~-40{deg}C and ~1 Pa) achievable on a long-duration balloon mission with a large detector payload. These characteristics enable the GAPS silicon tracker system to identify cosmic antinuclei via a novel technique based on exotic atom formation, de-excitation, and annihilation. Production and large-scale calibration of ~1000 detectors has begun for the first GAPS flight, scheduled for late 2021. The detectors developed for GAPS may also have other applications, for example in heavy nuclei identification.