Background: The evolution of shell structure around doubly magic exotic nuclei is of great interest in nuclear physics and astrophysics. In the `southwest region of $^{78}$Ni, the development of deformation might trigger a major shift in our understanding of explosive nucleosynthesis. To this end, new spectroscopic information on key close-lying nuclei is very valuable. Purpose: We intend to measure the isomeric and $beta$ decay of $^{75}$Co, with one-proton and two-neutron holes relative to $^{78}$Ni, to access new nuclear structure information in $^{75}$Co and its $beta$-decay daughters $^{75}$Ni and $^{74}$Ni. Methods: The nucleus $^{75}$Co is produced in relativistic in-flight fission reactions of $^{238}$U at the Radioactive Ion Beam Factory in the RIKEN Nishina Center. Its isomeric and $beta$ decay are studied exploiting the BigRIPS and EURICA setups. Results: We obtain partial $beta$-decay spectra for $^{75}$Ni and $^{74}$Ni, and report a new isomeric transition in $^{75}$Co. The energy [$E_{gamma}=1914(2)$ keV] and half-life [$t_{1/2}=13(6)$ $mu$s] of the delayed $gamma$ ray lend support for the existence of a $J^{pi}=(1/2^-)$ isomeric state at 1914(2) keV. A comparison with PFSDG-U shell-model calculations provides a good account for the observed states in $^{75}$Ni, but the first calculated $1/2^-$ level in $^{75}$Co, a prolate $K=1/2$ state, is predicted about 1 MeV below the observed $(1/2^-)$ level. Conclusions: The spherical-like structure of the lowest-lying excited states in $^{75}$Ni is proved. In the case of $^{75}$Co, the results suggest that the dominance of the spherical configurations over the deformed ones might be stronger than expected below $^{78}$Ni. Further experimental efforts to discern the nature of the $J^{pi}=(1/2^-)$ isomer are necessary.
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