  In this paper, we establish the relationship between backward stochastic
Volterra integral equations (BSVIEs, for short) and a kind of non-local
quasilinear (and possibly degenerate) parabolic equations. We first introduce
the extended backward stochastic Volterra integral equations (EBSVIEs, for
short). Under some mild conditions, we establish the well-posedness of EBSVIEs
and obtain some regularity results of the adapted solution to the EBSVIEs via
Malliavin calculus. We show that a given function expressed in terms of the
solution to the EBSVIEs solves a certain system of non-local parabolic partial
differential equations (PDEs, for short), which generalizes the famous
nonlinear Feynman-Kac formula in Pardoux{Peng [21].
