  Let $G$ be a group. Two elements $x,y \in G$ are said to be in the same
$z$-class if their centralizers in $G$ are conjugate within $G$. Consider
$\mathbb F$ a perfect field of characteristic $\neq 2$, which has a non-trivial
Galois automorphism of order $2$. Further, suppose that the fixed field
$\mathbb F_0$ has the property that it has only finitely many field extensions
of any finite degree. In this paper, we prove that the number of $z$-classes in
the unitary group over such fields is finite. Further, we count the number of
$z$-classes in the finite unitary group $U_n(q)$, and prove that this number is
same as that of $GL_n(q)$ when $q>n$.
