Results on ϕ meson production in inelastic p+ p collisions at CERN SPS energies are presented. They are derived from data collected by the NA61/SHINE fixed target experiment, by means of invariant mass spectra fits in the ϕ→ K⁺K^- decay channel. They include the first ever measured double differential spectra of ϕ mesons as a function of rapidity y and transverse momentum p_T for proton beam momenta of 80GeV/c and 158GeV/c, as well as single differential spectra of y or p_T for beam momentum of 40GeV/c. The corresponding total ϕ yields per inelastic p+ p event are obtained. These results are compared with existing data on ϕ meson production in p+ p collisions. The comparison shows consistency but superior accuracy of the present measurements. The emission of ϕ mesons in p+ p reactions is confronted with that occurring in Pb + Pb collisions, and the experimental results are compared with model predictions. It appears that none of the considered models can properly describe all the experimental observations.