  We propose a novel class of degenerate higher-order scalar-tensor theories as
an extension of mimetic gravity. By performing a noninvertible conformal
transformation on "seed" scalar-tensor theories which may be nondegenerate, we
can generate a large class of theories with at most three physical degrees of
freedom. We identify a general seed theory for which this is possible.
Cosmological perturbations in these extended mimetic theories are also studied.
It is shown that either of tensor or scalar perturbations is plagued with
gradient instabilities, except for a special case where the scalar
perturbations are presumably strongly coupled, or otherwise there appear ghost
instabilities.
