  Braided embeddings are embeddings to a product disc bundle so the projection
to the first factor is a branched cover. In this paper, we study which branched
covers lift to braided embeddings, which is a generalization of the Borsuk-Ulam
problem. We determine when a braided embedding in the complement of branch
locus can be extended over the branch locus in smoothly (or locally flat
piecewise linearly), and use it in conjunction with Hansen's criterion for
lifting covers. We show that every branched cover over an orientable surface
lifts to a codimension two braided embedding in the piecewise linear category,
but there are non-liftable branched coverings in the smooth category. We
explore the liftability question for covers over the Klein bottle. In dimension
three, we consider simple branched coverings over the three sphere, branched
over two-bridge, torus and pretzel knots, obtaining infinite families of
examples where the coverings do and do not lift. Finally, we also discuss some
examples of non-liftable branched covers in higher dimensions.
