The first experimental evidence for the existence of image-potential states in carbon nanotubes is presented. The observed features constitute a new class of surface image states due to their quantized centrifugal motion. Measurements of binding energies and the temporal evolution of image state electrons were performed using femtosecond time-resolved photoemission. The associated lifetimes are found to be significantly longer than those of n=1 image state on graphite, indicating a substantial difference in electron decay dynamics between tubular and planar graphene sheets.