We study the behavior of Lipschitz functions on intrinsic $C^1$ submanifolds of Heisenberg groups: our main result is their almost everywhere tangential Pansu differentiability. We also provide two applications: a Lusin-type approximation of Lipschitz functions on $HH$-rectifiable sets, and a coarea formula on $HH$-rectifiable sets that completes the program started in~cite{JNGV}.
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