  We define derived versions of $F$-zips and associate a derived $F$-zip to any
proper, smooth morphism of schemes in positive characteristic. We analyze the
stack of derived $F$-zips and certain substacks. We make a connection to the
classical theory and look at problems that arise when trying to generalize the
theory to derived $G$-zips and derived $F$-zips associated to lci morphisms. As
an application, we look at Enriques-surfaces and analyze the geometry of the
moduli stack of Enriques-surfaces via the associated derived $F$-zips. As there
are Enriques-surfaces in characteristic $2$ with non-degenerate Hodge-de Rham
spectral sequence, this gives a new approach, which could previously not be
obtained by the classical theory of $F$-zips.
