  This paper introduces and studies the analogue of the notion of Lipschitz
differentiability space by Cheeger, using Carnot groups and Pansu derivatives
as models. We call such metric measure spaces Pansu differentiability spaces
(PDS). After fixing a Carnot group $\mathbb G$, we prove three main results.
  (i) Being a PDS with $\mathbb G$-valued charts is equivalent to having
$\mathrm{rank}(\mathbb G)$ independent and horizontally universal Alberti
representations with respect to complete $\mathbb G$-valued charts. This result
leverages on a characterization by D. Bate, and extends it to our setting. For
non-Abelian Carnot groups, the completeness assumption cannot be removed as in
the Euclidean case. One direction of this equivalence can be seen as a metric
analogue of Pansu-Rademacher theorem.
  (ii) In every PDS the push-forward of the measure with respect to every
$\mathbb G$-valued chart is absolutely continuous with respect to the Haar
measure on $\mathbb G$. This extends the proof of Cheeger's conjecture by De
Philippis-Marchese-Rindler to our setting.
  (iii) For $Q$ being the homogeneous dimension of $\mathbb G$, being a PDS
with $\mathbb G$-valued charts, with finite $Q$-upper density, and positive
$Q$-lower density almost everywhere, is equivalent to being $\mathbb
G$-biLipschitz rectifiable. This extends a result by D. Bate and S. Li to our
setting. Moreover, the previous equivalence is false if we substitute $\mathbb
G$-biLipschitz with $\mathbb G$-Lipschitz, contrarily to what happens in the
Euclidean realm.
