  We construct a $p$-adic Rankin-Selberg $L$-function associated to the product
of two families of modular forms, where the first is an ordinary (Hida) family,
and the second an arbitrary universal-deformation family (without any
ordinarity condition at $p$). This gives a function on a 4-dimensional base
space - strictly larger than the ordinary eigenvariety, which is 3-dimensional
in this case. We prove our $p$-adic $L$-function interpolates all critical
values of the Rankin-Selberg $L$-functions for the classical specialisations of
our family, and derive a functional equation for our $p$-adic $L$-function.
