The study of the properties of disks around young brown dwarfs can provide important clues on the formation of these very low-mass objects and on the possibility of forming planetary systems around them. The presence of warm dusty disks around brown dwarfs is well known, based on near- and mid-infrared studies. High angular resolution observations of the cold outer disk are limited; we used ALMA to attempt a first survey of young brown dwarfs in the $ρ$-Oph star-forming region. All 17 young brown dwarfs in our sample were observed at 890 $μ $m in the continuum at $∼0.\!^{\prime\prime}5$ angular resolution. The sensitivity of our observations was chosen to detect $∼0.5$ M$_⊕$ of dust. We detect continuum emission in 11 disks ($∼65$\%\ of the total), and the estimated mass of dust in the detected disks ranges from $∼0.5$ to $∼6$ M$_⊕$. These disk masses imply that planet formation around brown dwarfs may be relatively rare and that the supra-Jupiter mass companions found around some brown dwarfs are probably the result of a binary system formation. We find evidence that the two brightest disks in $ρ$-Oph have sharp outer edges at R