The new edition of SGA3 by Philippe Gille et Patrick Polo provides a connected example due to Raynaud. It is in SGA3 Exposé VIA Exemples 1.3.2 (2) and you may find it at http://www.math.jussieu.fr/~polo/SGA3/Exp6A-23mai11.pdf.
This example is $2$-dimensional, but it is easy to modify it to get a $1$-dimensional example if the characteristic of the base field is at least $3$. More precisely, let $G$ be defined by the equations $X^p-tY^p=Y^p-tZ^p=0$ in the additive group $\mathbb{G}_a^3$ over the field $\mathbb{F}_p(t)$, $p\neq 2$. Then $G$ is a connected group scheme of dimension $1$ whose reduction is not a subgroup scheme, as may be seen by following step by step the arguments of loc. cit.
