From the preview page of Garnir's Dream Spaces with Hamel Bases, by Norbert Brunner [Arch. Math. Logik 26 (1987), 123-126]: "The situation changes completely, if one drops AC and instead assumes e.g. the axiom determinacy AD plus the principle of dependent choices DC. Then every linear map $A:X \to Y$ from a Banach space $X$ into a normed space $Y$ is continuous." (Brunner then says that Garnir establishes this from a weaker hypothesis.) Also, Wikipedia tells me that the axiom of dependent choice implies countable choice, and countable choice implies that every infinite set has a countably infinite subset. If $\ell^\infty$ (or any other infinite-dimensional Banach space $X$) has a Hamel basis, this contradicts the above setup. A Hamel basis for an infinite-dimensional Banach space $X$ is infinite, and assuming a countably infinite subset, you can make an unbounded and therefore discontinuous linear map from $X$ to $\mathbb{R}$.
(This topic is not my business at all, but maybe I am right as a naive student.)