  We generalize the classical construction principles of infinite-dimensional
real (and complex) Lie groups to the case of Lie groups over non-discrete
topological fields. In particular, we discuss linear Lie groups, mapping
groups, test function groups, diffeomorphism groups, and weak direct products
of Lie groups. The specific tools of differential calculus required for the Lie
group constructions are developed. Notably, we establish differentiability
properties of composition and evaluation, as well as exponential laws for
function spaces. We also present techniques to deal with the subtle
differentiability and continuity properties of non-linear mappings between
spaces of test functions. Most of the results are independent of any specific
properties of the topological vector spaces involved; in particular, we can
deal with real and complex Lie groups modeled on non-locally convex spaces.
