I would personally say “long-term student.” “Old” student implies that that person no longer studies with you but did in the past. For example: “Natalia is one of my old students. I told her 10 years ago.”

I would refer to that student as a "long-term student". If I were speaking more casually, I would say that student is an "old regular". For example: Chan is currently one of my long-term students. We've been working together for 5 years. I've worked with Adam for 5 years now. They're an old regular of mine.

Long-term student sounds better. Old student could refer to their age too 🙌