I have read on different occasions that unramified simple Rapoport-Zink spaces are formally smooth, eg. stated in Remark 4.13 of Rapoport and Viehman's survey article. These spaces are formal schemes $\mathcal M$ over the formal spectrum $\mathrm{Spf}(\mathcal O_E)$ of the ring of integers of an unramified extension $E/\mathbb Q_p$. By general theory, they are also locally formally of finite type.
I have found at least two definitions of smoothness for morphisms of formal schemes. The first says that it is a combination of being formally smooth and locally formally of finite type. Thus, by this definition my Rapoport-Zink space $\mathcal M$ is smooth over $\mathrm{Spf}(\mathcal O_E)$.
A second definition says that a morphism of formal schemes $f:\mathfrak X \to \mathfrak Y$ is smooth if, when we describe both $\mathfrak X$ and $\mathfrak Y$ as inductive systems $(\mathfrak X_n)$ and $(\mathfrak Y_n)$ via compatible choices of ideals of definition, all the morphisms of schemes $f_n:\mathfrak X_n \to \mathfrak Y_n$ are smooth. Then, let me take $I$ the largest ideal of definition of $\mathcal M$. For $n\geq 1$ the scheme $\mathcal M_n = (\mathcal M,\mathcal O_{\mathcal M}/I^n)$ should be smooth over $\mathcal O_{E}/p^n\mathcal O_{E}$. For $n=1$ it is the reduced special fiber $\mathcal M_1=\mathcal M_{\mathrm{red}}$, thus a smooth scheme over the residue field $\kappa(E)$.
This is all fine, until I read Vollaard and Wedhorn's paper where the geometry of the special fiber of the PEL simple unramified unitary Rapoport-Zink space of signature $(1,n-1)$ is described. Their Rapoport-Zink space is denoted by $\mathcal N$. Even though it is not stated in the paper, by the above I ought to expect $\mathcal N$ to be smooth. However, among other things, their geometric description implies that the reduced special fiber $\mathcal N_{\mathrm{red}}$ is in fact... not smooth! The smooth locus is determined to be the maximal Ekedahl-Oort stratum of $\mathcal N_{\mathrm{red}}$.
As far as I can tell, the moduli space considered by Vollaard and Wedhorn really is in the setting of Rapoport and Viehman's "unramified simple PEL Rapoport-Zink spaces". This looks silly, but I can't point out from what technicality the apparent contradiction stems. Would somebody see where the confusion lies ?
