Stellar activity is ubiquitously encountered in M dwarfs and often characterised by the Hα line. In the most active M dwarfs, Hα is found in emission, sometimes with a complex line profile. Previous studies have reported extended wings and asymmetries in the Hα line during flares. We used a total of 473 high-resolution spectra of 28 active M dwarfs obtained by the CARMENES (Calar Alto high-Resolution search for M dwarfs with Exo-Earths with Near-infrared and optical Echelle Spectrographs) spectrograph to study the occurrence of broadened and asymmetric Hα line profiles and their association with flares, and examine possible physical explanations. We detected a total of 41 flares and 67 broad, potentially asymmetric, wings in Hα. The broadened Hα lines display a variety of profiles with symmetric cases and both red and blue asymmetries. Although some of these line profiles are found during flares, the majority are at least not obviously associated with flaring. We propose a mechanism similar to coronal rain or chromospheric downward condensations as a cause for the observed red asymmetries; the symmetric cases may also be caused by Stark broadening. We suggest that blue asymmetries are associated with rising material, and our results are consistent with a prevalence of blue asymmetries during the flare onset. Besides the Hα asymmetries, we find some cases of additional line asymmetries in He I D3, Na I D lines, and the He I line at 10 830 Å taken all simultaneously thanks to the large wavelength coverage of CARMENES. Our study shows that asymmetric Hα lines are a rather common phenomenon in M dwarfs and need to be studied in more detail to obtain a better understanding of the atmospheric dynamics in these objects.