Let me try a possible answer. Take a model of $ZF$ where the axiom of choice for a denumerable family of finite sets holds but where there is an infinite Dedekind finite set $B$ (this model can be checked to exist, for instance, [here][1]; $\mathcal{M}32$ is one such model). Then $\ell_2(B)$ is an infinite dimensional Hilbert space with a Dedekind finite orthonormal base, whose unit ball is, by theorem 2 of Brunner's 1983 article _[Sequential compactness and the axiom of choice][2]_, sequentially compact.


  [1]: https://web.archive.org/web/20171102221007/http://consequences.emich.edu:80/conseq.htm
  [2]: https://doi.org/10.1305/ndjfl/1093870222