A search for the processes of double beta decay of $^{82}$Se into excited states of $^{82}$Kr is conducted using the OBELIX setup. The measurements are conducted using a 600 cm$^{3}$ HPGe detector. The detector is surrounded by 6.5 kg of enriched selenium (the content of $^{82}$Se was ~92.5%). The data collected over 17 572.2 h of measurements are given. As a result, a world-record sensitivity is reached for transitions of this type. Specifically, we have found an indication of the existence of two-neutrino $^{82}$Se decay into the first 0$^{+}^{82}$Kr excited state with a half-life of $4.4_{{ - 0.9}}^{{ + 1.6}}$ × 10$^{22}$ years.