  Warped cones are metric spaces introduced by John Roe from discrete group
actions on compact metric spaces to produce interesting examples in coarse
geometry. We show that a certain class of warped cones $\mathcal{O}_\Gamma (M)$
admit a fibred coarse embedding into a $L_p$-space ($1\leq p<\infty$) if and
only if the discrete group $\Gamma$ admits a proper affine isometric action on
a $L_p$-space. This actually holds for any class of Banach spaces stable under
taking Lebesgue-Bochner $L_p$-spaces and ultraproducts, e.g., uniformly convex
Banach spaces or Banach spaces with nontrivial type. It follows that the
maximal coarse Baum-Connes conjecture and the coarse Novikov conjecture hold
for a certain class of warped cones which do not coarsely embed into any
$L_p$-space for any $1\leq p<\infty$.
