  We construct differential operators for families of overconvergent Hilbert
modular forms by interpolating the Gauss--Manin connection on strict
neighborhoods of the ordinary locus. This is related to work done by Harron and
Xiao and by Andreatta and Iovita in the case of modular forms and by Zheng Liu
for Siegel modular forms. It has applications in particular to $p$-adic
$L$-functions of CM fields.
