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We are interested in the harmonic analysis on $p$-adic homogeneous spacesbased on spherical functions. In the present paper, we investigate the space$X$ of unitary hermitian matrices of odd size over a ${\mathfrak p}$-adic fieldof odd residual characteristic, which is a continuation of our previous paperwhere we have studied for even size matrices. First we give the explicitrepresentatives of the Cartan decomposition of $X$ and introduce a typicalspherical function $\omega(x;z)$ on $X$. After studying the functionalequations, we give an explicit formula for $\omega(x;z)$, where Hall-Littlewoodpolynomials of type $C_n$ appear as a main term, though the unitary groupacting on $X$ is of type $BC_n$. By spherical transform, we show the Schwartzspace ${\mathcal S}(K \backslash X)$ 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 giveparametrization of all the spherical functions on $X$ and the explicitPlancherel formula on ${\mathcal S}(K \backslash X)$.

Spherical functions on the space of $p$-adic unitary hermitian matrices II, the case of odd size