  The ${\cal N}{=}8, 1D$ analytic bi-harmonic superspace is shown to provide a
natural setting for ${\cal N}{=}8$ supersymmetric mechanics associated with the
off-shell multiplet ${\bf (4, 8, 4)}$ . The latter is described by an analytic
superfield $q^{1,1}$, and we construct the general superfield and component
actions for any number of such multiplets. The set of transformations
preserving the flat superspace constraints on $q^{1,1}$ constitutes ${\cal
N}{=}8$ extension of the two-dimensional Heisenberg algebra {\bf h}(2), with an
operator central charge. The corresponding invariant $q^{1,1}$ action is
constructed. It is unique and breaks 1D scale invariance. We also find a
one-parameter family of scale-invariant $q^{1,1}$ actions which, however, are
not invariant under the full ${\cal N}{=}8$ Heisenberg supergroup. Based on
preserving the bi-harmonic Grassmann analyticity, we formulate ${\cal N}{=}8,
1D$ supergravity in terms of the appropriate analytic supervielbeins. For its
truncated version we construct, both at the superfield and component levels,
the first example of off-shell $q^{1,1}$ action with local ${\cal N}{=}8, 1D$
supersymmetry. This construction can be generalized to any number of
self-interacting $q^{1,1}$ .
