The question I am going to ask is attributed to Antonio Avilés; see Question 7.7 in:

* J. Garbulińska and W. Kubiś, [Remarks on Gurarii spaces][1], *Extracta Math.* **26** (2011), 235–269.

however I learnt that only having already asked it myself.

> Can one construct in ZFC a Banach space of [density character][2] $\omega_1$ that does not have an equivalent *strictly convex* norm?

Maybe one may apply some kind of a Löwenheim–Skolem-type argument to a space that does not have a strictly convex norm?

Notes:

* every separable space has an equivalent strictly convex norm;
* the classical examples of spaces without a strictly convex norm include $\ell_\infty / c_0$ (see also [this post][3]) and $\ell_\infty(\Gamma)$ for any uncountable set $\Gamma$.


  [1]: https://arxiv.org/abs/1111.5840
  [2]: https://www.encyclopediaofmath.org/index.php/Density_(of_a_topological_space)
  [3]: https://mathoverflow.net/a/151956/15129