Aims.We present a search for low-mass brown dwarfs in the Pleiades open cluster. The identification of Pleiades members fainter and cooler than those currently known allows us to constrain evolutionary models for L dwarfs and to extend the study of the cluster mass function to lower masses. Methods: .We conducted a 1.8 deg2 near-infrared J-band survey at the 3.5 m Calar Alto Telescope, with completeness J_cpl˜ 19.0. The detected sources were correlated with those of previously available optical I-band images (I_cpl˜ 22). Using a J versus I-J colour-magnitude diagram, we identified 18 faint red L-type candidates, with magnitudes 17.4 3.2. If Pleiades members, their masses would span ~0.040-0.020~M&sun;. We performed follow-up HK_s-band imaging to further confirm their cluster membership by photometry and proper motion. Results: .Out of 11 IJ candidates with proper motion measurements, we find six cluster members, two non-members and three whose membership is uncertain and depends on the intrinsic velocity dispersion of Pleiades brown dwarfs. This dispersion (>4 mas yr-1) is at least four times that of cluster stars with masses ⪆1 M&sun;. Five of the seven other IJ candidates are discarded because their J-Ks colours are bluer than those of confirmed members. Our least massive proper motion members are BRB 28 and 29 (~25 M_Jup). The J versus I-J sequence of the L-type candidates at J>18 is not as red as theoretical models predict; it rather follows the field L-dwarf sequence translated to the cluster distance. This sequence overlapping, also observed in the J versus J-H and J-K diagrams, suggests that Pleiades and field L dwarfs may have similar spectral energy distributions and luminosities, and thus possibly similar radii. Also, we find α= 0.5±0.2 for a power-law approximation dN/dM ∝ M-α of the survey mass spectrum in the mass range 0.5-0.026~M&sun;. This value is similar to that of much younger clusters, indicating no significant differential evaporation of low-mass Pleiades members relative to more massive ones.