In this paper, we study the cohomology of the unitary unramified PEL Rapoport-Zink space of signature $(1,n-1)$ at maximal level. Our method revolves around the spectral sequence associated to the open cover by the analytical tubes of the closed Bruhat-Tits strata in the special fiber, which were constructed by Vollaard and Wedhorn. The cohomology of these strata, which are isomorphic to generalized Deligne-Lusztig varieties, has been computed in arXiv:2110.00614 [math.RT]. This spectral sequence allows us to prove the semisimplicity of the Frobenius action and to describe the inertial supports of the irreducible subquotients in the individual cohomology groups. In particular, when $n\geq 5$ no such subquotient is supercuspidal, but when $1\leq n \leq 4$ supercuspidal subquotients do occur. Moreover, we also prove that the cohomology groups need not be admissible in general. Via $p$-adic uniformization, we relate the cohomology of the Rapoport-Zink space to the cohomology of the basic stratum of a Shimura variety with no level at $p$. In the case $n=3$ or $4$, we give a complete description of the cohomology of the basic stratum in terms of automorphic representations. In particular, some automorphic representations occur with multiplicities dependent on $p$.