  In this paper we investigate partial spreads of $H(2n-1,q^2)$ through the
related notion of partial spread sets of hermitian matrices, and the more
general notion of constant rank-distance sets. We prove a tight upper bound on
the maximum size of a linear constant rank-distance set of hermitian matrices
over finite fields, and as a consequence prove the maximality of extensions of
symplectic semifield spreads as partial spreads of $H(2n-1,q^2)$. We prove
upper bounds for constant rank-distance sets for even rank, construct large
examples of these, and construct maximal partial spreads of $H(3,q^2)$ for a
range of sizes.
