  We consider the local model of a Shimura variety of PEL type, with the
unitary similitudes corresponding to a ramified quadratic extension of
$\mathbb{Q}_p$ as defining group. We examine the cases where the level
structure at $p$ is given by a parahoric that is the stabilizer of a selfdual
periodic lattice chain and that is special in the sense of Bruhat--Tits theory.
We prove that in these cases the special fiber of the local model is
irreducible and generically reduced; consequently, the special fiber is reduced
and is normal, Frobenius split, and with only rational singularities. In
addition, we show that in these cases the local model contains an open subset
that is isomorphic to affine space.
