Extension of relative rigid homomorphisms from the formal multiplicative group

A theorem of L\"utkebohmert states that a rigid group homomorphism from the formal multiplicative group to a smooth commutative rigid group $G$, with relatively compact image, can be extended to a homomorphism from the rigid multiplicative group to $G$. In this paper, we prove a relative version of this theorem over a geometrically reduced quasi-compact quasi-separated rigid space. The relative theorem is proved under an additional hypothesis that some open relative subgroup of $G$ has good reduction. This theorem is useful for studying rigid uniformisation of abelian or semiabelian varieties in a relative setting.