An example of a differentiability space which is PI-unrectifiable

Abstract: 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.