We have observed the transversal vibration mode of suspended carbon nanotubes at millikelvin temperatures by measuring the single-electron tunneling current. The suspended nanotubes are actuated contact-free by the radio frequency electric field of a nearby antenna; the mechanical resonance is detected in the time-averaged current through the nanotube. Sharp, gate-tunable resonances due to the bending mode of the nanotube are observed, combining resonance frequencies of up to nu(0) = 350 MHz with quality factors above Q = 10(5), much higher than previously reported results on suspended carbon nanotube resonators. The measured magnitude and temperature dependence of the Q factor shows a remarkable agreement with the intrinsic damping predicted for a suspended carbon nanotube. By adjusting the radio frequency power on the antenna, we find that the nanotube resonator can easily be driven into the nonlinear regime.