TL;DR
Yes. If you don't live in France, you can set up a company and be its president without a French residence permit, even if you are not an EU citizen: your Italian, Spanish or German residence permit is not required for this. What does require a French permit is settling in France or actually performing the role there on a regular basis.
What you need to know
- Your residence permit in another EU country does not give you the right to work in France: stays in France remain limited to 90 days in any 180.
- For identity, your passport is enough; there is no need to translate your foreign residence permit. Sources (2)
Service-publicService-public: can a foreign national set up a business in France?Service-publicService-public: how can a foreign national run a company in France? - The bank may ask for proof of address in your country of residence.
With LegalRapide
At LegalRapide, applications from founders of other nationalities are reviewed case by case, because opening the bank account and the registered office address depend on our partners' checks.
Written by Razvan Toma, founder of LegalRapide. How we check our figures · LinkedIn