  We are interested in the harmonic analysis on $p$-adic homogeneous spaces
based on spherical functions. In the present paper, we investigate the space
$X$ of unitary hermitian matrices of odd size over a ${\mathfrak p}$-adic field
of odd residual characteristic, which is a continuation of our previous paper
where we have studied for even size matrices. First we give the explicit
representatives of the Cartan decomposition of $X$ and introduce a typical
spherical function $\omega(x;z)$ on $X$. After studying the functional
equations, we give an explicit formula for $\omega(x;z)$, where Hall-Littlewood
polynomials of type $C_n$ appear as a main term, though the unitary group
acting on $X$ is of type $BC_n$. By spherical transform, we show the Schwartz
space ${\mathcal S}(K \backslash X)$ is a free Hecke algebra ${\mathcal H}(G,
K)$-module of rank $2^n$, where $2n+1$ is the size of matrices in $X$, and give
parametrization of all the spherical functions on $X$ and the explicit
Plancherel formula on ${\mathcal S}(K \backslash X)$.
