We present new and accurate near-infrared (NIR) J, K-band time series data for the Galactic globular cluster (GC) M5 = NGC5904. Data were collected with SOFI at the NTT (71 J + 120 K images) and with NICS at the TNG (25 J + 22 K images) and cover two orthogonal strips across the centre of the cluster of ≈ 5 x 10 arcmin² each. These data allowed us to derive accurate mean K-band magnitudes for 52 fundamental (RR(ab)) and 24 first overtone (RR(c)) RR Lyrae stars. Using this sample of RR Lyrae stars, we find that the slope of the K-band period-luminosity (PLK) relation (-2.33 ± 0.08) agrees quite well with similar estimates available in the literature. We also find, using both theoretical and empirical calibrations of the PLK relation, a true distance to M5 of 14.44 ± 0.02 mag. This distance modulus agrees very well (1 σ) with distances based on main-sequence fitting method and on kinematic method (14.44 ± 0.41 mag), while it is systematically smaller than the distance based on the white dwarf cooling sequence (14.67 ± 0.18 mag), even if with a difference slightly larger than 1 σ. The true distance modulus to M5 based on the J-band period-luminosity relation (14.50 ± 0.08 mag) is in quite good agreement with the distance based on the PLK relation, further supporting the use of NIR PL relations for RR Lyrae stars to improve the precision of the GC distance scale. ; peer reviewed: yes ; NRC Pub: yes