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The Lott-Sturm-Villani curvature-dimension condition $\mathsf{CD}(K,N)$provides a synthetic notion for a metric measure space to have curvaturebounded from below by $K$ and dimension bounded from above by $N$. It has beenrecently proved that this condition does not hold in sub-Riemannian geometryfor every choice of the parameters $K$ and $N$. In this paper, we extend thisresult to the context sub-Finsler geometry, showing that the $\mathsf{CD}(K,N)$condition is not well-suited to characterize curvature in this setting.Firstly, we show that this condition fails in (strict) sub-Finsler manifoldsequipped with a smooth strongly convex norm and with a positive smooth measure.Secondly, we focus on the sub-Finsler Heisenberg group, proving thatcurvature-dimension bounds can not hold also when the reference norm is lessregular, in particular when it is of class $C^{1,1}$. The strategy for provingthese results is a non-trivial adaptation of the work of Juillet [Rev. Mat.Iberoam., 37(1):177-188, 2021], and it requires the introduction of new toolsand ideas of independent interest. Finally, we demonstrate the failure of the(weaker) measure contraction property $\mathsf{MCP}(K,N)$ in the sub-FinslerHeisenberg group, equipped with a singular strictly convex norm and with apositive smooth measure. This result contrasts with what happens in the \srHeisenberg group, which instead satisfies $\mathsf{MCP}(0,5)$.

Failure of the curvature-dimension condition in sub-Finsler manifolds