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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 size $m$ over a ${\mathfrak p}$-adic field$k$ and give unified description including dyadic case, which is a continuationof our previous papers on non-dyadic case. The space becomes complicated when$e = v_\pi(2) > 0$. First we introduce a typical spherical function$\omega(x;z)$ on $X$, and study their functional equations, which depend on $m$and $e$, we give an explicit formula for $\omega(x;z)$, where Hall-Littlewoodpolynomials of type $C_n$ appear as a main term with different specializationaccording as $m = 2n$ or $2n+1$, but independent of $e$. By sphericaltransform, we show the Schwartz space ${\mathcal S}(K \backslash X)$ is a freeHecke algebra ${\mathcal H}(G,K)$-module of rank $2^n$, and giveparametrization of all the spherical functions on $X$ and the explicitPlancherel formula on ${\mathcal S}(K \backslash X)$. The Plancherel measuredoes not depend on $e$, but the normalization of $G$-invariant measure on $X$depends.

Harmonic analysis on the space of $p$-adic unitary hermitian matrices, including dyadic case