Semisimple symplectic characters of finite unitary groups

Let $G=\mathrm{U}(2m,{\mathbb{F}}_{q^2})$ 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 $\mathrm{U}(n,{\mathbb{F}}_{q^2})$ at any central element, using the characteristic map of the finite unitary group.