We present ALMA observations of the 2M1207 system, a young binary made of a brown dwarf with a planetary-mass companion at a projected separation of about 40 au. We detect emission from dust continuum at 0.89 mm and from the J = 3 - 2 rotational transition of CO from a very compact disk around the young brown dwarf. The small radius found for this brown dwarf disk may be due to truncation from the tidal interaction with the planetary-mass companion. Under the assumption of optically thin dust emission, we estimate. a dust mass of 0.1 M-circle plus. for the 2M1207A disk and a 3 sigma upper limit of similar to 1 M-Moon for dust surrounding 2M1207b, which is the tightest upper limit obtained so far for the mass of dust particles surrounding a young planetary-mass companion. We discuss the impact of this and other non-detections of young planetary-mass companions for models of planet formation that predict circumplanetary material to surround these objects.