  We prove that several classical Banach space properties are equivalent to
separability for the class of Lipschitz-free spaces, including Corson's
property ($\mathcal{C}$), Talponen's Countable Separation Property, or being a
G\^ateaux differentiability space. On the other hand, we single out more
general properties where this equivalence fails. In particular, the question
whether the duals of non-separable Lipschitz-free spaces have a weak$^*$
sequentially compact ball is undecidable in ZFC. Finally, we provide an example
of a nonseparable dual Lipschitz-free space that fails the Radon-Nikod\'ym
property.
