  We extend the construction of the $p$-adic $L$-function interpolating unitary
Friedberg--Jacquet periods in arXiv:2110.05426 to include the $p$-adic
variation of Maass--Shimura differential operators. In particular, we develop a
theory of nearly overconvergent automorphic forms in higher degrees of coherent
cohomology for unitary Shimura varieties generalising the results in
arXiv:2311.14438. The construction of this $p$-adic $L$-function can be viewed
as a higher-dimensional generalisation of the work of
Bertolini--Darmon--Prasanna and Castella--Hsieh, and the inclusion of this
extra variable arising from the $p$-adic iteration of differential operators
will play a key role in relating values of this $p$-adic $L$-function to
$p$-adic regulators of special cycles on unitary Shimura varieties.
