Detecting light at the single-photon level is one of the pillars of emergent photonic technologies. This is realized through state-of-the-art superconducting detectors that offer efficient, broadband and fast response. However, the use of low T_C superconducting thin films limits their operation temperature to approximately 4 K and below. Here, we demonstrate proof-of-concept nanodetectors based on exfoliated, two-dimensional cuprate superconductor Bi₂Sr₂CaCu₂O_8-δ that exhibit single-photon sensitivity at telecom wavelength at a record temperature of T = 20 K. These non-optimized devices exhibit a slow (∼ ms) reset time and a low detection efficiency (∼10⁻⁴). We realize the elusive prospect of single-photon sensitivity on a high-T_C nanodetector thanks to a novel approach, combining van der Waals fabrication techniques and a non-invasive nanopatterning based on light ion irradiation. This result paves the way for broader application of single-photon technologies, relaxing the cryogenic constraints for single-photon detection at telecom wavelength.