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

In 2022, Hatori gave a sufficient condition for complex Banach spaces to have the complex Mazur–Ulam property. In this paper, we introduce a class of complex Banach spaces $B$ that do not satisfy the condition but enjoy the property that every surjective isometry on the unit sphere of such $B$ admits an extension to a surjective real linear isometry on the whole space $B$. Typical examples of Banach spaces studied in this note are the spaces $\mathrm{L}\mathrm{i}\mathrm{p}([0,1])$ 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 $\mathrm{\mid }f(0)\mathrm{\mid }+\mathrm{\parallel }{f}^{\mathrm{\prime }}{\mathrm{\parallel }}_{\mathrm{\infty }}$.