  In the setting of fractional minimal surfaces, we prove that if two nonlocal
minimal sets are one included in the other and share a common boundary point,
then they must necessarily coincide.
  This strict maximum principle is not obvious, since the surfaces may touch at
an irregular point, therefore a suitable blow-up analysis must be combined with
a bespoke regularity theory to obtain this result.
  For the classical case, an analogous result was proved by Leon Simon. Our
proof also relies on a Harnack Inequality for nonlocal minimal surfaces that
has been recently introduced by Xavier Cabr\'e and Matteo Cozzi and which can
be seen as a fractional counterpart of a classical result by Enrico Bombieri
and Enrico Giusti.
  In our setting, an additional difficulty comes from the analysis of the
corresponding nonlocal integral equation on a hypersurface, which presents a
remainder whose sign and fine properties need to be carefully addressed.
