This talk is about joint work with Eyal Goren. Let E be a quadratic imaginary field and S a unitary Shimura variety associated with E. If p is a good inert prime, the special fiber of S at p carries an interesting p-foliation in its tangent bundle. It is defined first on the dense open -ordinary stratum, where it is shown to be related to Moonen's generalized Serre-Tate coordinates. We study its extension to deeper Ekedahl-Oort strata. We identify the quotient by the foliation with the e'tale component of a unitary Shimura variety with parahoric level structure at p, and deduce some information about the structure of the e'tale and multiplicative components of the latter. Finally we study integral subvarieties of the foliation.