In this paper we analyze the peculiar radio structure observed across the central region of the galaxy cluster Abell 585 (z=0.12). In the low-resolution radio maps, this structure appears uniform and diffuse on angular scales of ~3 arcmin, and is seemingly related to the distant (z=2.5) radio quasar B3 0727+409 rather than to the cluster itself. However, after a careful investigation of the unpublished archival radio data with better angular resolution, we resolve the structure into two distinct arcmin-scale features, which resemble typical lobes of cluster radio galaxies with no obvious connection to the background quasar. We support this conclusion by examining the spectral and polarization properties of the features, demonstrating in addition that the analyzed structure can hardly be associated with any sort of a radio mini-halo or relics of the cluster. Yet at the same time we are not able to identify host galaxies of the radio lobes in the available optical and infrared surveys. We consider some speculative explanations for our findings, including gravitational wave recoil kicks of SMBHs responsible for the lobes formation in the process of merging massive ellipticals within the central parts of a rich cluster environment, but we do not reach any robust conclusions regarding the origin of the detected radio features.