The Herschel open-time key program Disc Emission via a Bias-free Reconnaissance in the Infrared and Sub-millimeter (DEBRIS) is an unbiased survey of the nearest 100 stars for each stellar type A-M observed with a uniform photometric sensitivity to search for cold debris disks around them. The analysis of the Photoconductor Array Camera and Spectrometer (PACS) photometric observations of the 94 DEBRIS M dwarfs of this program is presented in this paper, following upon two companion papers on the DEBRIS A-star and FGK-star subsamples. In the M-dwarf subsample, two debris disks have been detected, around the M3V dwarf GJ 581 and the M4V dwarf Fomalhaut C (LP 876-10). This result gives a disk detection rate of 2.1^+2.7^_-0.7_% at the 68% confidence level, significantly less than measured for earlier stellar types in the DEBRIS program. However, we show that the survey of the DEBRIS M-dwarf subsample is about ten times shallower than the surveys of the DEBRIS FGK subsamples when studied in the physical parameter space of the disk's fractional dust luminosity versus blackbody radius. Furthermore, had the DEBRIS K-star subsample been observed at the same shallower depth in this parameter space, its measured disk detection rate would have been statistically consistent with the one found for the M-dwarf subsample. Hence, the incidence of debris disks does not appear to drop from the K subsample to the M subsample of the DEBRIS program, when considering disks in the same region of physical parameter space. An alternative explanation is that the only two bright disks discovered in the M-dwarf subsample would not, in fact, be statistically representative of the whole population.