Is it possible to add a facet to the faceted navigation / filters to filter the items on the PLP by inventory location or would a extension be needed?

Haven't tried it though but technically it should work. Just add the field in facet and try it.

Did add it, the facet shows up but the array of values is empty (I.e. no locations show up)

Which default field did you use?

location

@Mohammad Sharaf Ali, Did you add the field to the search fieldset?

@Kearobi it should be added under Facet Fields