It is well known that some lattices in ${\rm SO}(n,1)$ can be
nontrivially deformed when included in ${\rm SO}(n+1,1)$ (e.g., via
bending on a totally geodesic hypersurface); this contrasts with the
(super) rigidity of higher rank lattices. M. Kapovich recently gave
the first examples of lattices in ${\rm SO}(3,1)$ which are locally
rigid in ${\rm SO}(4,1)$ by considering closed hyperbolic
$3$-manifolds obtained by Dehn filling on hyperbolic two-bridge
knots. We generalize this result to Dehn filling on a more general
class of one-cusped finite volume hyperbolic $3$-manifolds, allowing
us to produce the first examples of closed hyperbolic $3$-manifolds
which contain embedded quasi-Fuchsian surfaces but are locally rigid
in ${\rm SO}(4,1)$.