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In 2022, Hatori gave a sufficient condition for complex Banach spaces to havethe complex Mazur--Ulam property. In this paper, we introduce a class ofcomplex Banach spaces $B$ that do not satisfy the condition but enjoy theproperty that every surjective isometry on the unit sphere of such $B$ admitsan extension to a surjective real linear isometry on the whole space $B$.Typical examples of Banach spaces studied in this note are the spaces ${\rmLip}([0,1])$ of all Lipschitz complex-valued functions on $[0,1]$ and$C^1([0,1])$ of all continuously differentiable complex-valued functions on$[0,1]$ equipped with the norm $|f(0)|+\|f'\|_\infty$.

Tingley's problem for complex Banach spaces which do not satisfy the Hausdorff distance condition