The low-spin structure of the semimagic ⁶⁴Ni nucleus has been considerably expanded: combining four experiments, several 0⁺ and 2⁺ excited states were identified below 4.5 MeV, and their properties established. The Monte Carlo shell model accounts for the results and unveils an unexpectedly complex landscape of coexisting shapes: a prolate 0⁺ excitation is located at a surprisingly high energy (3463 keV), with a collective 2⁺ state 286 keV above it, the first such observation in Ni isotopes. The evolution in excitation energy of the prolate minimum across the neutron N = 40 subshell gap highlights the impact of the monopole interaction and its variation in strength with N.