Exterior powers of π-divisible modules over fields

Let O be the ring of integers of a non-Archimedean local field and π a fixed uniformizer of O. We prove that the exterior powers of a π-divisible O-module scheme of dimension at most 1 over a field exist and commute with field extensions. We calculate the height and the dimension of the exterior powers in terms of the height of the given π-divisible O-module scheme.