A characterisation of the Daugavet property in spaces of vector-valued Lipschitz functions Medina Sabino, Rubén Rueda Zoca, Abraham Lipschitz-free space Tensor product Daugavet property Octahedral norms Perturbation of Lipschitz maps This work was supported by MCIN/AEI/10.13039/501100011033: Grant PID2021-122126NB-C31 and by Junta de Andalucía: Grants FQM-0185 and PY20_00255. The research of Rubén Medina was also supported by FPU19/04085 MIU (Spain) Grant, by GA23-04776S project (Czech Republic) and by SGS22/053/OHK3/1T/13 project (Czech Republic). The research of Abraham Rueda Zoca was also supported by Fundación Séneca: ACyT Región de Murcia grant 21955/PI/22 and by Generalitat Valenciana project CIGE/2022/97. Let M be a metric space and X be a Banach space. In this paper we address several questions about the structure of F(M )̂ ⊗π X and Lip0(M, X). Our results are the following: (1) We prove that if M is a length metric space then Lip0(M, X) has the Daugavet property. As a consequence, if M is length we obtain that F(M )̂ ⊗π X has the Daugavet property. This gives an affirmative answer to [13, Question 1] (also asked in [24, Remark 3.8]). (2) We prove that if M is a non-uniformly discrete metric space or an unbounded metric space then the norm of F(M )̂ ⊗π X is octahe- dral, which solves [6, Question 3.2 (1)]. (3) We characterise all the Banach spaces X such that L(X, Y ) is octahedral for every Banach space Y , which solves a question by Johann Langemets. 2025-04-04T12:11:53Z 2025-04-04T12:11:53Z 2023-10-26 journal article Published version: Medina Sabino, Rubén y Rueda Zoca, Abraham. Journal of Functional Analysis Volume 289, Issue 1, 1 July 2025, 110896. https://doi.org/10.1016/j.jfa.2025.110896 https://hdl.handle.net/10481/103462 10.1016/j.jfa.2025.110896 10.48550/arXiv.2305.05956 eng http://creativecommons.org/licenses/by-nc-nd/4.0/ open access Attribution-NonCommercial-NoDerivatives 4.0 Internacional Elsevier