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It is known that if $M$ is a finite-dimensional Banach space, or a strictlyconvex space, or the space $\ell_1$, then every non-expansive bijection $F: B_M\to B_M$ is an isometry. We extend these results to non-expansive bijections$F: B_E \to B_M$ between unit balls of two different Banach spaces. Namely, if$E$ is an arbitrary Banach space and $M$ is finite-dimensional or strictlyconvex, or the space $\ell_1$ then every non-expansive bijection $F: B_E \toB_M$ is an isometry.

Non-expansive bijections between unit balls of Banach spaces (A technical version with some boring proofs included)