  Let G be a Lie group over a local field of positive characteristic which
admits a contractive automorphism f (i.e., the forward iterates f^n(x) of each
group element x converge to the neutral element 1). We show that then G is a
torsion group of finite exponent and nilpotent. We also obtain results
concerning the interplay between contractive automorphisms of Lie groups over
local fields, contractive automorphisms of their Lie algebras, and positive
gradations thereon. Some of the results even extend to Lie groups over
arbitrary complete ultrametric fields.
