  Let $F$ be a totally real field and let $E/F$ be a CM quadratic extension. We
construct a $p$-adic $L$-function attached to Hida families for the group
$GL_{2/F} \times GL_{1/E}$. It interpolates critical Rankin--Selberg $L$-values
at all classical points corresponding to representations $\pi\boxtimes \chi$
with the weights of $\chi$ smaller than the weights of $\pi$. This confirms a
conjecture of the author, making related results on the $p$-adic Beilinson
conjecture and its Iwasawa-theoretic variants unconditional.
  Our $p$-adic $L$-function agrees with a previous result of Hida when $E/F$
splits above $p$, and it is new otherwise. We build it as a ratio of families
of global and local Waldspurger zeta integrals, the latter constructed using
the local Langlands correspondence in families.
  In an appendix of possibly independent recreational interest, we give a
reality-TV-inspired proof of an identity concerning double factorials.
