Assume that 19 > 2, and let Omicron(K) be a p-adic discrete valuation ring with residue field admitting a finite p-basis, and let R be a formally smooth formally finite-type Omicron(K)-algebra. (Indeed, we allow slightly more general rings R.) We construct an antiequivalence of categories between the categories of p-divisible groups over R and certain semi-linear algebra objects which generalize (phi,(sic))-modules of height s 1 (or Kisin modules). A similar classification result for p-power order finite flat group schemes is deduced from the classification of p-divisible groups. We also show compatibility of various construction of (Zp-lattice or torsion) Galois representations, including the relative version of Faltings' integral comparison theorem for p-divisible groups. We obtain partial results when p=2.