The NA61/SHINE experiment at the CERN Super Proton Synchrotron (SPS) studies the onset of deconfinement in hadron matter by a scan of particle production in collisions of nuclei with various sizes at a set of energies covering the SPS energy range. This paper presents results on inclusive double-differential spectra, transverse momentum and rapidity distributions and mean multiplicities of $\pi ^\pm $, $K^\pm $, p and $\bar{p}$ produced in the 20% most central$^7$Be+$^9$Be collisions at beam momenta of 19A, 30A, 40A, 75A and 150A ${\mathrm{Ge} \mathrm{V}}\!/\!c$. The energy dependence of the $K^\pm $/$\pi ^\pm $ ratios as well as of inverse slope parameters of the $K^\pm $ transverse mass distributions are close to those found in inelastic p+p reactions. The new results are compared to the world data on p+p and Pb+Pb collisions as well as to predictions of the Epos, Urqmd, Ampt, Phsd and Smash models.