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Let $p$ be a prime. Let $R$ be a regular local ring of dimension $d\ge 2$whose completion is isomorphic to $C(k)[[x_1,\ldots,x_d]]/(h)$, with $C(k)$ aCohen ring with the same residue field $k$ as $R$ and with $h\inC(k)[[x_1,\ldots,x_d]]$ such that its reduction modulo $p$ does not belong tothe ideal $(x_1^p,\ldots,x_d^p)+(x_1,\ldots,x_d)^{2p-2}$ of$k[[x_1,\ldots,x_d]]$. We extend a result of Vasiu-Zink (for $d=2$) to showthat each Barsotti-Tate group over $\text{Frac}(R)$ which extends to everylocal ring of $\text{Spec}(R)$ of dimension $1$, extends uniquely to aBarsotti-Tate group over $R$. This result corrects in many cases several errorsin the literature. As an application, we get that if $Y$ is a regular integralscheme such that the completion of each local ring of $Y$ of residuecharacteristic $p$ is a formal power series ring over some complete discretevaluation ring of absolute ramification index $e\le p-1$, then eachBarsotti-Tate group over the generic point of $Y$ which extends to every localring of $Y$ of dimension $1$, extends uniquely to a Barsotti-Tate group over$Y$.

Purity for Barsotti-Tate groups in some mixed characteristic situations