The bright exciton emission of carbon nanotubes is appealing for optoelectronic devices and fundamental studies of light-matter interaction in one-dimensional nanostructures. However, to date, the photophysics of excitons in carbon nanotubes is largely affected by extrinsic effects. Here we perform time-resolved photoluminescence measurements over 14 orders of magnitude for ultra-clean carbon nanotubes bridging an air gap over pillar posts. Our measurements demonstrate a new regime of intrinsic exciton photophysics with prolonged spontaneous emission times up to T1=18 ns, about two orders of magnitude better than prior measurements and in agreement with values hypothesized by theorists about a decade ago. Furthermore, we establish for the first time exciton decoherence times of individual nanotubes in the time domain and find fourfold prolonged values up to T2=2.1 ps compared with ensemble measurements. These first observations motivate new discussions about the magnitude of the intrinsic dephasing mechanism while the prolonged exciton dynamics is promising for applications.