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Let $n$ be a positive integer and $A^{\prime }$ a complete characteristic zero discrete valuation ring with maximal ideal $𝔪$, absolute ramification index $e\<p-1$ and perfect residue field $k$ of characteristic $p\>2$. In this paper we classify smooth finite dimensional formal $p$- groups over $A_n^{\prime }=A^{\prime }/{𝔪}^n{A}^{\prime }$, groups on which the “multiplication by $p$” morphism is faithfully flat, in particular $p$-divisible groups. As applications, we prove that $p$-divisible groups over $k$, and the morphisms between them, lift canonically to $A^{\prime }/p{A}^{\prime }$, and we study liftings...