Recall that a Banach space $X$ has the \textit{Daugavet property} if, given any rank-one linear and continuous operator $T:X\longrightarrow X$ it follows that $$\Vert T+\mathrm{I}\Vert=1+\Vert T\Vert,$$ where $\mathrm{I}:X\longrightarrow X$ stands for the identity operator. In the context of the Lipschitz free space over a complete metric space, it is known that $\mathcal F(M)$ has the Daugavet property if, and only if, the metric space $M$ is \textit{length}, i.e., if for every pair of distinct points $x,y\in M$ then $d(x,y)$ equals the inf. of the length of all the rectifiable curves joininig $x$ and $y$ [1] and if, and only if, its dual $\Lip(M)$ has the Daugavet property. In this talk we will analyse the question when $\Lip(M,X)$ has the Daugavet property for every Banach space $X$. The results of this talk are part of the preprint \cite{mr} in collaboration with R. Medina [2].
[1] L. García-Lirola, A. Procházka and A. Rueda Zoca, A characterisation of the Daugavet property in spaces of Lipschitz functions, J. Math. Anal. Appl. 464 (2018), 473–492.
[2] R. Medina and A. Rueda Zoca, A characterisation of the Daugavet property in spaces of vector-valued Lipschitz functions, preprint. Available at ArXiV.org with reference arXiv:2305.05956.