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, 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 $\omega_1$ 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) and $\ell_\infty(\Gamma)$ for any uncountable set $\Gamma$.