In this work we present the results of a direct imaging survey for brown dwarf companions around the nearest stars at the mid-infrared 10 micron range (λ c = 8.7 μm, Δλ = 1.1 μm) using the CanariCam instrument on the 10.4 m Gran Telescopio Canarias (GTC). We imaged the 25 nearest stellar systems within 5 pc of the Sun at declinations δ > -25° (at least half have planets from radial-velocity studies), reaching a mean detection limit of 11.3 ± 0.2 mag (1.5 mJy) in the Si-2 8.7 μm 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 50 au, with effective temperatures down to 600 K and masses greater than 30 M Jup assuming an average age of 5 Gyr and masses down to the deuterium-burning mass limit for objects with ages <1 Gyr. 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 ≳ 30 M Jup and T eff ≳ 600 K at ~3.5-35 au projected orbital separations.