The Turdu salt mine in Romania certainly seems to have some enormous underground chambers.
In terms of slate, Aberllefenni, Hen Gloddfa, Wrysgan all have enormous underground cavities, there are also slate mines in Luxembourg (now mostly flooded) with large, almost vertical chambers.
I've been to Kiruna but didn't get a chance to go down the mine, I haven't seen any pictures of enormous caverns, I have seen many of large workings but none of them with a huge vertical distance. The iron mines around Bergslagen are large too, some of the chambers in those mines are huge, Tuna Hastberg comes to mind
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