We have performed continuum-discretized coupled channels (CDCC) calculations of the breakup of $^{8}\mathrm{B}$ on $^{58}\mathrm{Ni}$ and direct proton transfer for the $^{8}\mathrm{B}+^{58}\mathrm{Ni}$ system at laboratory energies of $20--28.4$ MeV. The influence of the $^{7}\mathrm{Be}$ core-target optical potential (OP) on the breakup cross section was investigated. Elastic scattering angular distributions for the $^{7}\mathrm{Be}+^{58}\mathrm{Ni}$ and $^{8}\mathrm{B}+^{58}\mathrm{Ni}$ systems at five different energies around the Coulomb barrier were studied, and a reasonable energy-independent OP for each system was obtained. Using these OPs and two different $^{7}\mathrm{Be}$-$p$ relative motion wave functions, and summing breakup and direct proton transfer contributions, we were able to fit the experimental cross section at a $^{8}\mathrm{B}$ laboratory energy of $25.75$ MeV. We calculated the excitation function for the $^{7}\mathrm{Be}$ emission in the $^{8}\mathrm{B}+^{58}\mathrm{Ni}$ reaction, where $^{7}\mathrm{Be}$ products were measured at the forward angle ${\ensuremath{\theta}}_{\mathrm{lab}}={45}^{\ifmmode^\circ\else\textdegree\fi{}}$ in the energy interval ${E}_{\mathrm{lab}}=20--28.4$ MeV. In view of the peripheral character of the $^{8}\mathrm{B}$ breakup reaction at near-barrier energies, we could extract the asymptotic normalization coefficient for the $^{7}\mathrm{Be}$-$p$ system, which was found to be ${C}_{\mathrm{Be}--p,{p}_{3/2}}^{2}=0.543\ifmmode\pm\else\textpm\fi{}0.027$ fm${}^{\ensuremath{-}1}$. Finally, the astrophysical ${S}_{17}$(0) factor was found to be ${S}_{17}(0)=20.8\ifmmode\pm\else\textpm\fi{}1.1$ eV b.