  Among all Poincar\'e inequality spaces, we define the class of Cheeger
fractals, which includes the sub-Riemannian Heisenberg group. We show that
there is no bi-Lipschitz embedding $\iota$ of any Cheeger fractal $X$ into any
Banach space $V$ with the following property: there exists a bounded Euclidean
domain $\Omega$ such that for any Lipschitz mapping $f \colon \Omega \to X$,
the Bochner partial derivatives of $\iota \circ f$ exist and are integrable.
This extends and provides context for an important related result of Creutz and
Evseev.
