If $A$ is a mixed characteristic complete DVR (I'm only actually interested in $\mathbf{Z}_p$) and $G/A$ is a closed subgroup scheme of $GL(n)$ whose generic fibre is connected reductive and split, is the identity component of the special fibre also connected reductive and split?

Oh dear I'm getting very confused about this question. Is there a closed subgroup scheme of $GL(2)$ over $A$ whose generic fibre is trivial but whose special fibre is a Borel subgroup of $GL(2)$? Is life really as bad as that?