On every infinite-dimensional Banach space there exists a discontinuous linear functional.
Solution 1:
No. There are models of $\mathsf{ZF+\lnot AC}$ in which every linear transformation from a Banach space to a normed space is automatically continuous. In particular this is true for linear functionals.
In such models, it follows, every linear functional has to be continuous.
An example for these models are Solovay's model, or models of $\mathsf{ZF+AD}$.