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Let $k$ be a perfect field of characteristic $p > 2$, and let $K$ be a finitetotally ramified extension of $W(k)[\frac{1}{p}]$ of ramification degree $e$.We consider an unramified base ring $R_0$ over $W(k)$ satisfying certainconditions, and let $R = R_0\otimes_{W(k)}\mathcal{O}_K$. Examples of such $R$include $R = \mathcal{O}_K[\![s_1, \ldots, s_d]\!]$ and $R =\mathcal{O}_K\langle t_1^{\pm 1}, \ldots, t_d^{\pm 1}\rangle$. We show that thegeneralization of Raynaud's theorem on extending $p$-divisible groups holdsover the base ring $R$ when $e < p-1$, whereas it does not hold when $R =\mathcal{O}_K[\![s]\!]$ with $e \geq p$. As an application, we prove that if$R$ has Krull dimension $2$ and $e < p-1$, then the locus of Barsotti-Taterepresentations of $\mathrm{Gal}(\overline{R}[\frac{1}{p}]/R[\frac{1}{p}])$cuts out a closed subscheme of the universal deformation scheme. If $R =\mathcal{O}_K[\![s]\!]$ with $e \geq p$, we prove that such a locus is not$p$-adically closed.

Extending $p$-divisible groups and Barsotti-Tate deformation ring in the relative case