We have carried out an intensive spectroscopic campaign to search for binaries via periodic radial velocity (RV) variations among all the nitrogen-rich WN Wolf-Rayet (WR) stars in the Small Magellanic Cloud (SMC), and all WNE stars in the Large Magellanic Cloud (LMC). We present in this first paper the results for the SMC. Along with the results of Bartzakos et al. on the only carbon/oxygen-rich WR star (AB8, WO4+O4), the whole WR population of the SMC (11 stars) has now been investigated intensively for periodic RV variability. We have also retrieved time-dependent photometric data in the public domain from the OGLE and MACHO projects, and X-ray data from the ROSAT and Chandra archives, to provide additional constraints on the binary character. Contrary to theoretical expectations that predict a virtually 100 per cent binary frequency in the SMC, we find a normal (∼40 per cent) WR binary frequency in this galaxy. We also find the clear presence of hydrogen in the winds of the single WR stars in the SMC, even for the stars with an early spectral subtype. We discuss the possible reasons and implications of this for stellar evolution of massive stars in such a low-metallicity environment, e.g. the influence of rotation versus the necessity of a very high initial mass of the progenitors for single stars, and the possible past occurrence of Roche lobe overflow for binaries.