We report on a sensitive infrared search for disks around isolated young planetary-mass objects (PMOs) in the NGC 1333 cluster, by stacking 70 Spitzer/IRAC frames at 3.6 and 4.5 μm. Our coadded images go >2.3 mag deeper than single-epoch frames, and cover 50 brown dwarfs, 15 of which have M9 or later spectral types. Spectral types >M9 correspond to masses in the giant-planet domain, i.e., near or below the deuterium-burning limit of 0.015 M⊙. Five of the 12 PMOs show definitive evidence of excess, implying a disk fraction of 42%, albeit with a large statistical uncertainty given the small sample. Comparing with measurements for higher-mass objects, the disk fraction does not decline substantially with decreasing mass in the substellar domain, consistent with previous findings. Thus, free-floating PMOs have the potential to form their own miniature planetary systems. We note that only one of the six lowest-mass objects in NGC 1333, with spectral type L0 or later, has a confirmed disk. Reviewing the literature, we find that the lowest-mass free-floating objects with firm disk detections have masses ∼0.01 M⊙ (or ∼10 MJup). It is not clear yet whether even lower-mass objects harbor disks. If not, it may indicate that ∼10 M Jup is the lower-mass limit for objects that form like stars. Our disk-detection experiment on deep Spitzer images paves the way for studies with JWST at longer wavelengths and higher sensitivity, which will further explore disk prevalence and formation of free-floating PMOs.