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No, NRIs generally don't need to obtain an Aadhaar card, and hence they can apply for a PAN card without an Aadhaar number.
If an NRI has already obtained an Aadhaar card or becomes a "resident" for Aadhaar purposes, meaning they stayed in India for more than 182 days in the last 12 months. Then, linking PAN card with their Aadhaar becomes mandatory to avoid your PAN card from becoming inoperative.
An NRI can apply for a PAN card via Protean (NSDL) or UTIITSL portals by filling out Form 49A for an Indian citizen or Form 49AA for a foreign citizen, providing passport and overseas/Indian address proofs instead of an Aadhaar.
Some alternatives for an Aadhaar card for an NRI PAN application are passport, visa, OCI/PIO card, overseas bank statement, or utility bill reflecting foreign address, and NRE/NRO statement or Indian address proof if using an Indian communication address.
If your PAN profile correctly reflects your "Non-Resident" status, you fall in the exempt category, and your PAN will remain active even without linking it with the Aadhaar. You will face problems when your PAN card is still tagged as "Resident".