  We construct a (Lipschitz) differentiability space which has at generic
points a disconnected tangent and thus does not contain positive measure
subsets isometric to positive measure subsets of spaces admitting a Poincar\'e
inequality. We also prove that $l^2$-valued Lipschitz maps are differentiable
a.e., but there are also Lipschitz maps taking values in some other Banach
spaces having the Radon-Nikodym property which fail to be differentiable on
sets of positive measure.
