Scalable quantum networks require quantum memories that are chip-integrated, telecom-band compatible, and capable of flexible retrieval. Nanofabricated mechanical resonators meet these criteria. They offer independent tunability of optical and mechanical modes, long-lived phonon states, and design flexibility beyond atomic systems, making them strong candidates for practical integrated quantum memory. Here, we demonstrate an on-chip, absorptive optomechanical memory for telecom-band photons, based on optomechanically induced transparency (OMIT) and operating near the mechanical ground state. The device stores telecom-band photons, demonstrating compatibility with external photon sources at the few-photon level, while enabling on-demand retrieval. By placing the device in a dilution refrigerator at 20 mK and tailoring the control field to suppress optical heating, we achieve a remarkably low phonon occupancy of just 0.32 during the storage process. Our results lay the groundwork for scalable, phonon-based quantum memory devices and open new avenues for integrating mechanical systems into practical quantum network architectures.