Please narrow down your question; do you want a specific car model, company, body style?
GM made convertibles from the early 1900s till about 1976, when they quit. I think the Cadillac Eldorado was the last one. Then, some years later they started up again. So, you could say that Cadillac, dating from 1904 or so, produced convertibles till 1976 continuously , except for the war years when civilian car production was suspended.
I’ll let others answer which specific car body/model had the longest production run. It could be the VW Beetle convertible or the Model T Ford, which was produced in the same body style for many years.
Hope that gives you a partial answer.
P.S. If you are looking for the convertible with th longest physical length, look in the Guinness Book of Records. I’m sure you can find it in your local library! In the 30s they built some very long Rolls Royce, Deusenberg and Auburn cars. They were all convertibles.