Aims: We search for low-mass companions of M dwarfs and characterize their multiplicity fraction with the purpose of helping in the selection of the most appropriate targets for the CARMENES exoplanet survey. Methods: We obtained high-resolution images in the I band with the lucky imaging instrument FastCam at the 1.5 m Telescopio Carlos Sánchez for 490 mid- to late-M dwarfs. For all the detected binaries, we measured angular separations, position angles, and magnitude differences in the I band. We also calculated the masses of each individual component and estimated orbital periods, using the available magnitude and colour relations for M dwarfs and our own MJ-spectral type and mass-MI relations. To avoid biases in our sample selection, we built a volume-limited sample of M0.0-M5.0 dwarfs that is complete up to 86% within 14 pc. Results: From the 490 observed stars, we detected 80 companions in 76 systems, of which 30 are new discoveries. Another six companion candidates require additional astrometry to confirm physical binding. The multiplicity fraction in our observed sample is 16.7 ± 2.0%. The bias-corrected multiplicity fraction in our volume-limited sample is 19.5 ± 2.3% for angular separations of 0.2 to 5.0 arcsec (1.4-65.6 au), with a peak in the distribution of the projected physical separations at 2.5-7.5 au. For M0.0-M3.5 V primaries, our search is sensitive to mass ratios higher than 0.3 and there is a higher density of pairs with mass ratios over 0.8 compared to those at lower mass ratios. Binaries with projected physical separations shorter than 50 au also tend to be of equal mass. For 26 of our systems, we estimated orbital periods shorter than 50 a, 10 of which are presented here for the first time. We measured variations in angular separation and position angle that are due to orbital motions in 17 of these systems. The contribution of binaries and multiples with angular separations shorter than 0.2 arcsec, longer than 5.0 arcsec, and of spectroscopic binaries identified from previous searches, although not complete, may increase the multiplicity fraction of M dwarfs in our volume-limited sample to at least 36%. Tables A.1-A.6 are only available at the CDS via anonymous ftp to http://cdsarc.u-strasbg.fr (http://130.79.128.5) or via http://cdsarc.u-strasbg.fr/viz-bin/qcat?J/A+A/597/A47