Manifolds are typically defined as follows: A topological space $M$ is called a manifold, if for all $x\in M$, there exists an open set $U \ni x$ such that $U$ is homeomorphic to $\mathbb{R}^n$, for some $n\in \mathbb{N}$.
One could replace $\mathbb{R}^n$ by another complete ordered (not necessarily Archimedean) field.
Schneider does so in his book $p$-Adic Lie Groups, developing the theory on manifolds over $\mathbb{Q}_p$ , and similar fields. Scheider makes this remark in the introduction:
The fundamental difference is that the $p$-adic notion has no geometric content. As we will see, a paracompact $p$-adic manifold is topologically a disjoint union of charts and therefore is, from a geometric standpoint, completely uninteresting.
Is there a reference developing the theory for manifolds/lie groups/lie algebras say over the hyper-reals $\mathbb{R}^*$, or Levi-Civita fields, or other various non-Archimedean fields?