Let $R$ be a discrete valuation ring with the field of fractions $K$ and the residue characteristic $p$. If $K$ is of characteristic $0$, then a celebrated theorem of Tate says that the pullback functor $G\mapsto G\_K$ is fully faithful on the category of $p$-divisible groups over $R$. I suppose that this full faithfulness fails if $K$ is instead of characteristic $p$; how can I see this failure?



The generic fiber pullback for $p$-divisible groups in characteristic $p$
The question of whether it is still fully faithful when the characteristic of $K$ is $p$ is mentioned in expose IX, SGA 7. The answer is yes, see "Homomorphisms of Barsotti-Tate groups and crystals in positive characteristic" by de Jong (Invent. math. 134, 301-333 (1998))


