  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$\nu$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.
