The first lithium-drifted silicon (Si(Li)) detectors to satisfy the unique geometric, performance, and cost requirements of the General Antiparticle Spectrometer (GAPS) experiment have been produced by Shimadzu Corporation. The GAPS Si(Li) detectors will form the first large-area, relatively high-temperature Si(Li) detector system with sensitivity to X-rays to operate at high altitude. These 10 cm-diameter, 2.5 mm-thick, 4- or 8-strip detectors provide the active area, X-ray absorption efficiency, energy resolution, and particle tracking capability necessary for the GAPS exotic-atom particle identification technique. In this paper, the detector performance is validated on the bases of X-ray energy resolution and reconstruction of cosmic minimum ionizing particle (MIP) signals. We use the established noise model for semiconductor detectors to distinguish sources of noise due to the detector from those due to signal processing electronics. We demonstrate that detectors with either 4 strips or 8 strips can provide the required $lesssim$4 keV (FWHM) X-ray energy resolution at flight temperatures of $-35$ to $-45^{circ}$C, given the proper choice of signal processing electronics. Approximately 1000 8-strip detectors will be used for the first GAPS Antarctic balloon flight, scheduled for late 2021.
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