In this work we present the results of a direct imaging survey for brown dwarf companions around the nearest stars at the mid-infrared 10um range ({lambda}_c_=8.7{mu}m, {Delta}{lambda}=1.1{mu}m) using the CanariCam instrument on the 10.4m Gran Telescopio Canarias (GTC). We imaged the 25 nearest stellar systems within 5pc of the Sun at declinations {delta}{>}-25{deg} (at least half have planets from radial-velocity studies), reaching a mean detection limit of 11.3+/-0.2mag (1.5mJy) in the Si-2 8.7{mu}m band over a range of angular separations from 1" to 10". This would have allowed us to uncover substellar companions at projected orbital separations between ~2 and 50au, with effective temperatures down to 600K and masses greater than 30M_Jup_ assuming an average age of 5Gyr and masses down to the deuterium-burning mass limit for objects with ages <1Gyr. From the nondetection of such companions, we determined upper limits on their occurrence rate at depths and orbital separations yet unexplored by deep imaging programs. For the M dwarfs, the main component of our sample, we found with a 90% confidence level that fewer than 20% of these low-mass stars have L- and T-type brown dwarf companions with m>~30M_Jup_ and T_eff_>~600K at ~3.5-35au projected orbital separations.