  We find the irreducible decomposition of the Weil representation of the
unitary group $\mathrm{U}_{2n}(A)$, where $A$ is a ramified quadratic extension
of a finite, commutative, local, principal ideal ring $R$ and the nilpotency
degree of the maximal ideal of $A$ is odd. We show in particular that this Weil
representation is multiplicity free. Restriction to the special unitary group
$\mathrm{SU}_{2n}(A)$ preserves irreducibility and multiplicity freeness
provided $n>1$.
