  The Floer cohomology and the Fukaya category are not defined in general.
Indeed, while the issue of obstructions can be theoretically addressed by
introducing bounding cochains, the actual existence of even one such bounding
cochain is usually unknown. This paper aims to deal with the problem. We study
certain non-archimedean analytic structure that enriches the Maurer-Cartan
theory of bounding cochains. Using this rigid structure and family Floer
techniques, we prove that within a connected family of graded Lagrangian
submanifolds, if any one Lagrangian is unobstructed (in a slightly stronger
sense), then all remaining Lagrangians are automatically unobstructed.
