Smooth Lie groups over local fields of positive characteristic need not be analytic

We describe finite-dimensional smooth Lie groups over local fields of positive characteristic which do not admit an analytic Lie group structure compatible with the given topological group structure, and Cn-Lie groups without a compatible Cn+1-Lie group structure for each n⩾1. We also present examples of non-analytic, smooth automorphisms of analytic Lie groups over such fields, as well as Cn-automorphisms which fail to be Cn+1.