Abstract The chemical abundance of r-process elements in nearby dwarf spheroidal (dSph) galaxies is a powerful tool to probe the site of r-process since their small-mass scale can assess the potential rarity of events associated with the r-process production. A merger of binary neutron stars is a promising candidate for such a site. In faint, or less-massive, dSph galaxies such as the Draco, a few binary neutron star mergers are expected to have occurred at most over the whole past. We have measured the chemical abundance, including Eu and Ba, of three red giants in the Draco dSph by Subaru High-Dispersion Spectrograph observation. The Eu detection for one star with [Fe/H] = −1.45 confirms a broadly constant [Eu/H] of ∼ −1.3 for stars with [Fe/H] ${\buildrel {>} \over \sim}-2$. This feature is shared by other dSphs with similar masses, i.e., the Sculptor and the Carina, and suggests that a neutron star merger is the origin of r-process elements in terms of the rarity of this event. In addition, two very metal-poor stars with [Fe/H] = −2.12 and −2.51 are found to exhibit very low Eu abundances, such as [Eu/H] < −2, with the suggestion of a sudden increase of Eu abundance by more than 0.7 dex at [Fe/H] ≈ −2.2 in the Draco dSph. The detection of Ba abundances for these stars suggests that the r-process enrichment began no later than the time when only a few percent of stars in the present-day Draco dSph were formed. Though identifying the origin of an early Eu production inside the Draco dSph should be left until more abundant data of stars with [Fe/H] ${\buildrel {<} \over \sim }$−2 in Draco, as well as other faint dSphs, become available, the implied early emergence of an Eu production event might be reconciled with the presence of extremely metal-poor stars enriched by r-process elements in the Galactic halo.