Thermal Wightman functions of a massless scalar field are studied within the framework of a `near horizon' static background model of an extremal R - N black hole. This model is built up by using global Carter-like coordinates over an infinite set of Bertotti - Robinson submanifolds glued together. The analytical extendibility beyond the horizon is imposed as constraints on (thermal) Wightman's functions defined on a Bertotti - Robinson submanifold. It turns out that only the Bertotti - Robinson vacuum state, i.e. T = 0, satisfies the above requirement. Furthermore, the extension of this state onto the whole manifold is proved to coincide exactly with the vacuum state in the global Carter-like coordinates. Hence a theorem similar to the Bisognano - Wichmann theorem for the Minkowski spacetime in terms of Wightman functions holds with vanishing `Unruh - Rindler temperature'. Furthermore, the Carter-like vacuum restricted to a Bertotti - Robinson region, resulting in a pure state there, has vanishing entropy despite the presence of event horizons. Some comments on the real extremal R - N black hole are given.