The LUCIFER project aims at deploying the first array of enriched scintillating bolometers for the investigation of neutrinoless double-beta decay of $^{82}$Se. The matrix which embeds the source is an array of ZnSe crystals, where enriched $^{82}$Se is used as decay isotope. The radiopurity of the initial components employed for manufacturing crystals, that can be operated as bolometers, is crucial for achieving a null background level in the region of interest for double-beta decay investigations. In this work, we evaluated the radioactive content in 2.5 kg of 96.3% enriched $^{82}$Se metal, measured with a high-purity germanium detector at the Gran Sasso deep underground laboratory. The limits on internal contaminations of primordial decay chain elements of $^{232}$Th, $^{238}$U and $^{235}$U are respectively: $<$61 $mu$Bq/kg, $< $110 $mu$Bq/kg and $<$74 $mu$Bq/kg at 90% C.L.. The extremely low-background conditions in which the measurement was carried out and the high radiopurity of the $^{82}$Se allowed us to establish the most stringent lower limits on the half-lives of double-beta decay of $^{82}$Se to 0$^+_1$, 2$^+_2$ and 2$^+_1$ excited states of $^{82}$Kr of 3.4$cdot$10$^{22}$ y, 1.3$cdot$10$^{22}$ y and 1.0$cdot$10$^{22}$ y, respectively, with a 90% C.L..
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