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Let $G$ be a group scheme over a scheme $S$ which is the spectrum of a discrete valuation ring. Let $\eta$ (resp. $s$) be the generic (resp. closed) point. Assume that the generic fiber $G_{\eta}$ is proper but $G_s^{0}$ is not and that $G_s$ has many connected components. Is there any contradiction/counterexample in finding a subscheme $H\subset G$ such that $H_{\eta}=G_{\eta}$, $G_s^0\subset H_s$ and $H_s$ proper (in particular, $H_s\subset G_s$)?

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Every semistable degeneration of an elliptic curve. –  Jason Starr Jul 17 '13 at 16:45
Since $G_s^0$ is closed in $G_s$, it is also closed in $H_s$ and therefore proper if $H_s$ is. –  Laurent Moret-Bailly Jul 17 '13 at 18:27

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