Context. Observations have shown that inner discs and rings (IDs and IRs) are not preferably found in barred galaxies, which indicates that their formation may differ from that described by the traditional bar-origin scenario in many cases. In contrast, the role of minor mergers in producing these inner components (ICs), while often invoked, is still poorly understood. Aims: We investigate the capability of minor mergers to trigger the formation of IDs and IRs in spiral galaxies through collisionless N-body simulations. Methods: We run a battery of minor merger simulations in which both primary and secondary galaxies are modelled as disc-bulge-halo galaxies with realistic density ratios. Different orbits and mass ratios are considered, as well as two different models for the primary galaxy (a Sab or Sc). We then perform a detailed analysis of the morphology, structure, and kinematics of the ICs resulting from the minor merger. Results: All the simulated minor mergers develop thin ICs out of satellite material, supported by rotation. A wide morphological zoo of ICs are obtained (including IDs, IRs, pseudo-rings, nested IDs, spiral patterns, and combinations of them), but all have structural and kinematical properties similar to those observed. The sizes of the resulting ICs are comparable to those observed in real galaxies with the adequate scaling. The existence of the resulting ICs can be deduced from the features that they imprint in the isophotal profiles and kinemetric maps of the final remnant, as in many real galaxies. Weak transitory oval distortions appear in the remnant centre in many cases, but none of them develops a noticeable bar. The realistic density ratios used in the present models ensure that the satellites experience more efficient orbital circularization and disruption than in previous studies. Combined with the disc resonances induced by the encounter, these processes produce highly aligned co- and counter-rotating ICs at the remnant centre. Conclusions: Minor mergers are an efficient mechanism for forming rotationally-supported stellar ICs in spiral galaxies, without requiring either strong dissipation or the development of noticeable bars. The present models indicate that minor mergers can account for the existence of pure-stellar old ICs in unbarred galaxies, and suggest that their role must have been crucial in the formation of ICs and much more complex than just bar triggering. Figures 12-16 are available in electronic form at http://www.aanda.org