Context. The star formation rate density (SFRD) evolution presents an area of great interest in the studies of galaxy evolution and reionization. The current constraints of SFRD at z >  5 are based on the rest-frame UV luminosity functions with the data from photometric surveys. The VIMOS UltraDeep Survey (VUDS) was designed to observe galaxies at redshifts up to ∼6 and opened a window for measuring SFRD at z >  5 from a spectroscopic sample with a well-controlled selection function.Aims. We establish a robust statistical description of the star-forming galaxy population at the end of cosmic HI reionization (5.0 ≤ z ≤ 6.6) from a large sample of 49 galaxies with spectroscopically confirmed redshifts. We determine the rest-frame UV and Lyα luminosity functions and use them to calculate SFRD at the median redshift of our sample z = 5.6.Methods. We selected a sample of galaxies at 5.0 ≤ zspec ≤ 6.6 from the VUDS. We cleaned our sample from low redshift interlopers using ancillary photometric data. We identified galaxies with Lyα either in absorption or in emission, at variance with most spectroscopic samples in the literature where Lyα emitters (LAE) dominate. We determined luminosity functions using the 1/Vmax method.Results. The galaxies in this redshift range exhibit a large range in their properties. A fraction of our sample shows strong Lyα emission, while another fraction shows Lyα in absorption. UV-continuum slopes vary with luminosity, with a large dispersion. We find that star-forming galaxies at these redshifts are distributed along the main sequence in the stellar mass vs. SFR plane, described with a slope α = 0.85 ± 0.05. We report a flat evolution of the specific SFR compared to lower redshift measurements. We find that the UV luminosity function is best reproduced by a double power law, while a fit with a Schechter function is only marginally inferior. The Lyα luminosity function is best fitted with a Schechter function. We derive a logSFRDUV(M⊙ yr−1 Mpc−3) = −1.45+0.06−0.08 and logSFRDLyα(M⊙ yr−1 Mpc−3) = −1.40+0.07−0.08. The SFRD derived from the Lyα luminosity function is in excellent agreement with the UV-derived SFRD after correcting for IGM absorption.Conclusions. Our new SFRD measurements at a mean redshift of z = 5.6 are ∼0.2 dex above the mean SFRD reported in Madau & Dickinson (2014, ARA&A, 52, 415), but in excellent agreement with results from Bouwens et al. (2015a, ApJ, 803, 34). These measurements confirm the steep decline of the SFRD at z >  2. The bright end of the Lyα luminosity function has a high number density, indicating a significant star formation activity concentrated in the brightest LAE at these redshifts. LAE with equivalent width EW > 25 Å contribute to about 75% of the total UV-derived SFRD. While our analysis favors low dust content in 5.0 <  z <  6.6, uncertainties on the dust extinction correction and associated degeneracy in spectral fitting will remain an issue, when estimating the total SFRD until future surveys extending spectroscopy to the NIR rest-frame spectral domain, such as with JWST.Key words: galaxies: high-redshift / galaxies: evolution / galaxies: formation / galaxies: star formation / dark ages, reionization, first stars / galaxies: luminosity function, mass function⋆ Based on data obtained with the European Southern Observatory Very Large Telescope, Paranal, Chile, under Large Program 185.A−0791.