  By $p$-adically interpolating the branching law for the spherical pair
$\left(U_n, U_{n+1} \times U_{n}\right)$ 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.
