Let $S$ be a scheme where $p$ is locally nilpotent and let $G$ be a $p$-divisible group over $S$.
Is connectedness of $G$ equivalent to $G[p] := \ker(p : G \to G) \to S$ radicial (universally injective)?
The motivation for this question is: in Messing's book "The Crystals Associated to Barsotti-Tate Groups", the second condition is equivalent to $G$ being equal to its completion $\overline{G}$, which is a formal Lie group (Proposition II.4.4 p.61), while in Tate's article "p-Divisible groups" the first condition is equivalent to $G$ being represented by a formal Lie group (Proposition 1 p.162).
Also in Messing's book the base $S$ is assumed to have $p$ locally nilpotent while in Tate's article this is not the case; could this assumption be removed?
