We report the detection of the radio afterglow of a long gamma-ray burst (GRB) 111005A at 5-345 GHz, including the very long baseline interferometry observations with the positional error of 0.2 mas. The afterglow position is coincident with the disk of a galaxy ESO 580-49 at z= 0.01326 (~1" from its center), which makes GRB 111005A the second closest GRB known to date, after GRB 980425. The radio afterglow of GRB 111005A was an order of magnitude less luminous than those of local low-luminosity GRBs, and obviously than those of cosmological GRBs. The radio flux was approximately constant and then experienced an unusually rapid decay a month after the GRB explosion. Similarly to only two other GRBs, we did not find the associated supernovae (SN), despite deep near- and mid-infrared observations 1-9 days after the GRB explosion, reaching ~20 times fainter than other SNe associated with GRBs. Moreover, we measured twice solar metallicity for the GRB location. The low gamma-ray and radio luminosities, rapid decay, lack of a SN, and super-solar metallicity suggest that GRB 111005A represents a different rare class of GRBs than typical core-collapse events. We modelled the spectral energy distribution of the GRB 111005A host finding that it is a dwarf, moderately star-forming galaxy, similar to the host of GRB 980425. The existence of two local GRBs in such galaxies is still consistent with the hypothesis that the GRB rate is proportional to the cosmic star formation rate (SFR) density, but suggests that the GRB rate is biased towards low SFRs. Using the far-infrared detection of ESO 580-49, we conclude that the hosts of both GRBs 111005A and 980425 exhibit lower dust content than what would be expected from their stellar masses and optical colours.