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It has no Kalifah, means it is not muslim.

My elders do not call Saudia an infidel country.
If they are asked then, in all probability, they will say that Saudia is indeed Dar-ul-Islam.
I have not asked but then you do not ask the obvious things.
I personally am not a fan of the current ideology prevalent in that region but even then I'll not call it Dar-ul-Kufr.
A Muslim is a person who accepts Islam.
Since overwhelming majority in Saudia will claim to be Muslim therefore we have to accept that it is a Muslim country.
Caliphate is Social Duty (Fardh-e-Kifaya). If a group does not do that then they all shall be sinners. This still does not take them out of the fold of Islam.
If you think otherwise, then I want you to study Islam further. W/salam You are my brother in Islam so it is a pleasure and honour to receive your suggestion. Can you suggest a specific document to study?
Wassalam