The distribution of multiplicity among low-mass stars is a key issue to understanding the formation of stars and brown dwarfs, and recent surveys have yielded large enough samples of nearby low-mass stars to study this issue statistically to good accuracy. Previously, we have presented a multiplicity study of ~700 early/mid M-type stars observed with the AstraLux high-resolution Lucky Imaging cameras. Here, we extend the study of multiplicity in M-type stars through studying 286 nearby mid/late M-type stars, bridging the gap between our previous study and multiplicity studies of brown dwarfs. Most of the targets have been observed more than once, allowing us to assess common proper motion to confirm companionship. We detect 68 confirmed or probable companions in 66 systems, of which 41 were previously undiscovered. Detections are made down to the resolution limit of ~100 mas of the instrument. The raw multiplicity in the AstraLux sensitivity range is 17.9%, leading to a total multiplicity fraction of 21%-27% depending on the mass ratio distribution, which is consistent with being flat down to mass ratios of ~0.4, but cannot be stringently constrained below this value. The semi-major axis distribution is well represented by a log-normal function with {mu}_a_= 0.78 and {sigma}_a_= 0.47, which is narrower and peaked at smaller separations than for a Sun-like sample. This is consistent with a steady decrease in average semi-major axis from the highest-mass binary stars to the brown dwarf binaries.