Is there a compact Hausdorff space that is hereditarily Lindelöf but non-metrizable? Under the continuum Hypothesis, such space exists (see the abstract of A compact L-space under CH by Kunen). I would what if we assume the opposite of the continuum Hypothesis.
From the abstract of Locally connected hereditarily Lindelöf compacta, it seems that such space exists with the negation of the continuum Hypothesis. But I don't understand the notation $MA$ and $CH$ therein.
The Double Arrow Space; questions like these may be answered by searching the pi-Base: https://topology.pi-base.org/spaces?q=Compact%2B%24T_2%24%2BHereditarily%20Lindel%C3%B6f%2B~Metrizable
In particular it's a compact separable linearly orderable topological space, and thus Hausdorff and hereditarily Lindelöf. But it's non-metrizable because it's not second-countable.