We report the discovery of a super-Earth orbiting the star GJ 536 based on the analysis of the radial-velocity time series from the HARPS and HARPS-N spectrographs. GJ 536 b is a planet with a minimum mass Msini of 5.36 ± 0.69 M⊕; it has an orbital period of 8.7076 ± 0.0025 d at a distance of 0.066610(13) AU, and an orbit that is consistent with circular. The host star is the moderately quiet M1 V star GJ 536, located at 10 pc from the Sun. We find the presence of a second signal at 43 d that we relate to stellar rotation after analysing the time series of Ca II H&K and Hα spectroscopic indicators and photometric data from the ASAS archive. We find no evidence linking the short period signal to any activity proxy. We also tentatively derived a stellar magnetic cycle of less than 3 yr. The data used in this paper (Table A.1) 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/A108