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We describe the structure of the supersingular locus of a Shimura variety fora quaternionic unitary similitude group of degree $2$ over a ramified odd prime$p$ if the level at $p$ is given by a special maximal compact open subgroup.More precisely, we show that such a locus has purely $2$-dimensional, and everyirreducible component is birational to the Fermat surface. Furthermore, we havean estimation of the numbers of connected and irreducible components. To provethese assertions, we completely determine the structure of the underlyingreduced scheme of the Rapoport--Zink space for the quaternionic unitarysimilitude group of degree $2$, with a special parahoric level. We prove thatsuch a scheme is purely $2$-dimensional, and every irreducible component isisomorphic to the Fermat surface. We also determine its connected components,irreducible components and their intersection behaviors by means of theBruhat--Tits building of $\mathrm{PGSp}_4(\mathbb{Q}_p)$. In addition, wecompute the intersection multiplicity of the GGP cycles associated to anembedding of the considering Rapoport--Zink space into the Rapoport--Zink spacefor the unramified $\mathrm{GU}_{2,2}$ with hyperspecial level for theminuscule case.

On supersingular loci of Shimura varieties for quaternionic unitary groups of degree $2$