  We establish the existence of smooth densities for solutions to a broad class
of path-dependent SDEs under a H\"ormander-type condition. The classical scheme
based on the reduced Malliavin matrix turns out to be unavailable in the
path-dependent context. We approach the problem by lifting the given
$n$-dimensional path-dependent SDE into a suitable $L_p$-type Banach space in
such a way that the lifted Banach-space-valued equation becomes a
state-dependent reformulation of the original SDE. We then formulate
H\"ormander's bracket condition in $\mathbb R^n$ for non-anticipative SDE
coefficients defining the Lie brackets in terms of vertical derivatives in the
sense of the functional It\^o calculus. Our pathway to the main result engages
an interplay between the analysis of SDEs in Banach spaces, Malliavin calculus,
and rough path techniques.
