  We construct new models of `curved' SU$(4|1)$ supersymmetric mechanics based
on two versions of the off-shell multiplet ${\bf(8,8,0)}$ which are `mirror' to
each other. The worldline realizations of the supergroup SU$(4|1)$ are treated
as a deformation of flat ${\cal N}{=}\,8$, $d\,{=}\,1$ supersymmetry. Using
SU$(4|1)$ chiral superfields, we derive invariant actions for the first-type
${\bf(8,8,0)}$ multiplet, which parametrizes special K\"ahler manifolds. Since
we are not aware of a manifestly SU$(4|1)$ covariant superfield formalism for
the second-type ${\bf(8,8,0)}$ multiplet, we perform a general construction of
SU$(4|1)$ invariant actions for both multiplet types in terms of SU$(2|1)$
superfields. An important class of such actions enjoys superconformal
OSp$(8|2)$ invariance. We also build off-shell actions for the SU$(4|1)$
multiplets ${\bf(6,8,2)}$ and ${\bf(7,8,1)}$ through appropriate substitutions
for the component fields in the ${\bf(8,8,0)}$ actions. The ${\bf(6,8,2)}$
actions are shown to respect superconformal SU$(4|1,1)$ invariance.
