  We present a simple family of Bell inequalities applicable to a scenario
involving arbitrarily many parties, each of which performs two binary-outcome
measurements. We show that these inequalities are members of the complete set
of full-correlation Bell inequalities discovered by
Werner-Wolf-Zukowski-Brukner. For scenarios involving a small number of
parties, we further verify that these inequalities are facet-defining for the
convex set of Bell-local correlations. Moreover, we show that the amount of
quantum violation of these inequalities naturally manifests the extent to which
the underlying system is genuinely many-body entangled. In other words, our
Bell inequalities, when supplemented with the appropriate quantum bounds,
naturally serve as device-independent witnesses for entanglement depth,
allowing one to certify genuine k-partite entanglement in an arbitrary $n\ge
k$-partite scenario without relying on any assumption about the measurements
being performed, nor the dimension of the underlying physical system. A brief
comparison is made between our witnesses and those based on some other Bell
inequalities, as well as the quantum Fisher information. A family of witnesses
for genuine k-partite nonlocality applicable to an arbitrary $n\ge k$-partite
scenario based on our Bell inequalities is also presented.
