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Let $F$ be a totally real field and let $E/F$ be a CM quadratic extension. Weconstruct a $p$-adic $L$-function attached to Hida families for the group ${\rmGL}_{2/F}\times {\rm Res}_{E/F}{\rm GL}_{1}$. It is characterised by an exactinterpolation property for critical Rankin-Selberg $L$-values, at classicalpoints corresponding to representations $\pi \boxtimes \chi$ with the weightsof $\chi$ smaller than the weights of$\pi$.  Our $p$-adic $L$-function agrees with previous results of Hida when $E/F$splits above $p$ or $F=\mathbf{Q}$, and it is new otherwise. Exploring a methodthat should bear further fruits, we build it as a ratio of families of globaland local Waldspurger zeta integrals, the latter constructed using the localLanglands correspondence in families. In an appendix of possibly independentrecreational interest, we give a reality-TV-inspired proof of an identityconcerning double factorials.

$p$-adic L-functions via local-global interpolation: the case of ${\rm GL}_2 \times {\rm GU}(1)$