Anticyclotomic $p$-adic $L$-functions for Coleman families of $U_{n+1}\times U_n$

By $p$-adically interpolating the branching law for the spherical pair $(U_n,U_{n+1}\times U_n)$ of definite unitary groups, we construct a $p$-adic $L$-function attached to cohomological automorphic representations of $U_{n+1}\times U_n$. Under a further multiplicity one assumption, we extend the construction to Coleman families. Our $p$-adic $L$-function interpolates the square root of the central critical $L$-value. It has weight and anticyclotomic variables and its construction relies on the proof of the unitary Gan–Gross–Prasad conjecture.