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We prove that in the Heisenberg group $\mathbb{H}^1$ with a sub-Finslerstructure, an $(X,Y)$-Lipschitz surface which is complete, oriented, connectedand stable must be a vertical plane. In particular, the result holds for entireintrinsic graphs of Euclidean Lipschitz functions.

The Bernstein problem for $(X,Y)$-Lipschitz surfaces in three-dimensional sub-Finsler Heisenberg groups