  Let $G = {\rm U}(2m, {\mathbb F}_{q^2})$ be the finite unitary group, with
$q$ the power of an odd prime $p$. We prove that the number of irreducible
complex characters of $G$ with degree not divisible by $p$ and with
Frobenius-Schur indicator -1 is $q^{m-1}$. We also obtain a combinatorial
formula for the value of any character of ${\rm U}(n, {\mathbb F}_{q^2})$ at
any central element, using the characteristic map of the finite unitary group.
