This paper discusses evidence for and properties of disks associated to brown dwarfs in the star-forming region $\rho~$Oph. We selected nine objects from the ISOCAM survey of Bontemps et al. (CITE) that have detections in the two mid-infrared bands (6.7 and 14.3 μm), relatively low extinction and low luminosity. We present low-resolution near-infrared spectra in the J, H and K bands, and determine for each source spectral type, extinction, effective temperature and luminosity by comparing the spectra to those of field dwarfs and to the most recent model stellar atmospheres. The results indicate that eight objects have spectral types M6–M7.5, effective temperature of 2600–2700 K, one has a later spectral type (M8.5) and lower temperature (about 2400 K). The derived extinctions range between $A_V\sim2$ and 8 mag. The location of the objects on the HR diagram, in spite of the uncertainties of the evolutionary tracks for young objects of substellar mass, indicates that all the objects are very young and have masses below about 0.08 $M_\odot$. The coolest object in our sample has mass in the range 8–12 MJ (0.008–0.012 $M_\odot$). In all cases, the mid-infrared excess is consistent with the predictions of models of disks irradiated by the central object, showing that circumstellar disks are commonly associated to young brown dwarfs and planetary-mass objects. Finally, we discuss possible variations of the disk geometry among different objects, as well as the possibility of using these data to discriminate between various formation scenarios.