  Let $ \CO $ be the ring of integers of a non-Archimedean local field and $
\pi $ a fixed uniformizer of $ \CO $. We establish three main results. The
first one states that the exterior powers of a $ \pi $-divisible $ \CO $-module
scheme of dimension at most 1 over a field exist and commute with algebraic
field extensions. The second one states that the exterior powers of a
$p$-divisible group of dimension at most 1 over arbitrary base exist and
commute with arbitrary base change. The third one states that when $ \CO $ has
characteristic zero, then the exterior powers of $ \pi $-divisible groups with
scalar $ \CO $-action and dimension at most 1 over a locally Noetherian base
scheme exist and commute with arbitrary base change. We also calculate the
height and dimension of the exterior powers in terms of the height of the given
$p$-divisible group or $ \pi $-divisible $ \CO $-module scheme.
