Primordial black holes (PBHs) hypothetically generated in the first instants of life of the Universe are potential dark matter (DM) candidates. Focusing on PBHs masses in the range $[5 times10^{14} - 5 times 10^{15}]$g, we point out that the neutrinos emitted by PBHs evaporation can interact through the coherent elastic neutrino nucleus scattering (CE$ u$NS) producing an observable signal in multi-ton DM direct detection experiments. We show that with the high exposures envisaged for the next-generation facilities, it will be possible to set bounds on the fraction of DM composed by PBHs improving the existing neutrino limits obtained with Super-Kamiokande. We also quantify to what extent a signal originating from a small fraction of DM in the form of PBHs would modify the so-called neutrino floor, the well-known barrier towards detection of weakly interacting massive particles (WIMPs) as the dominant DM component.