Complete metric spaces with property (Z) are length metric spaces

We prove that every complete metric space with property (Z) is a length space. These answers questions posed by Garc'{i}a-Lirola, Proch'{a}zka and Rueda Zoca, and by Becerra Guerrero, L'{o}pez-P'{e}rez and Rueda Zoca, related to the structure of Lipschitz-free Banach spaces of metric spaces.