Context. M dwarfs are known to generate the strongest magnetic fields among main-sequence stars with convective envelopes, but we are still lacking a consistent picture of the link between the magnetic fields and underlying dynamo mechanisms, rotation, and activity. Aims: In this work we aim to measure magnetic fields from the high-resolution near-infrared spectra taken with the CARMENES radial-velocity planet survey in a sample of 29 active M dwarfs and compare our results against stellar parameters. Methods: We used the state-of-the-art radiative transfer code to measure total magnetic flux densities from the Zeeman broadening of spectral lines and filling factors. Results: We detect strong kG magnetic fields in all our targets. In 16 stars the magnetic fields were measured for the first time. Our measurements are consistent with the magnetic field saturation in stars with rotation periods P < 4 d. The analysis of the magnetic filling factors reveal two different patterns of either very smooth distribution or a more patchy one, which can be connected to the dynamo state of the stars and/or stellar mass. Conclusions: Our measurements extend the list of M dwarfs with strong surface magnetic fields. They also allow us to better constrain the interplay between the magnetic energy, stellar rotation, and underlying dynamo action. The high spectral resolution and observations at near-infrared wavelengths are the beneficial capabilities of the CARMENES instrument that allow us to address important questions about the stellar magnetism.