Primarily motivated by the similarities between the underdoped superconducting cuprates and the granular systems in regards of electric conductivity, phase fluctuations of the order parameter, and nuclear spin-lattice relaxation, a study has been carried out in a NbN(111) textured film at controlled granularity by means of superconducting quantum interference device magnetization and $^{93}\text{N}\text{b}$ NMR measurements. The Meissner diamagnetism in zero-field-cooling and field-cooling conditions and for different orientation of the magnetic field and the isothermal magnetization curves around the superconducting transition temperature ${T}_{\text{c}}$, are studied. $^{93}\text{N}\text{b}$ spectra and relaxation measurements have been performed for two values of the external magnetic field in parallel and perpendicular geometry, in the temperature range 4--300 K. In the superconducting phase the experimental findings for the textured film are similar to the one in bulk NbN. The nuclear spin-lattice relaxation process is the same as in bulk NbN in the temperature range 50--300 K, confirming a dominant contribution to the density of states at the Fermi energy arising from the Nb $4d$ band. At variance, on cooling from about 40 K down to ${T}_{\text{c}}$ $(H)$, the $^{93}\text{N}\text{b}$ relaxation rate in the film dramatically departs from the expected behavior for the Fermi gas and mimics the opening of a spin gap. The interpretation of the spin-gap opening in terms of depletion in the density of states at the Fermi energy can justify the anomalous temperature behavior of the $^{93}\text{N}\text{b}$ relaxation rate on approaching ${T}_{\text{c}}$ $(H)$ from above. The experimental findings suggest the occurrence of superconducting fluctuations (density-of-states term) in one-dimensional regime, coupled to a reduction in the time of flight of the electrons, both effects being related to the granularity. The data also suggest that the spin-gap phase in underdoped cuprates could be connected more to granularity, rather than to exotic mechanisms of magnetic origin.