We demonstrate multi-gigahertz continuous-wave mode-locking of a Yb:KLuW waveguide laser. A femtosecond-laser-inscribed Yb:KLuW channel waveguide in an extended laser cavity delivers a fundamentally mode-locked laser near 1030 nm. A tunable few-centimeter-long cavity containing a single-walled carbon nanotube saturable absorber as mode-locker generates self-starting femtosecond pulses with average output powers exceeding 210 mW at repetition rates of 2.27, 2.69, and 3.55 GHz. The laser cavity, which includes a wedged waveguide, is extended by using a lens pair that controls the laser fluence on the saturable absorber for reliable mode-locked operation without instability. The presented laser performance, mode-locked up to 3.55 GHz, highly suggests the potential of crystalline Yb:KLuW waveguides for realizing high-power ultrafast lasers with higher GHz repetition rates in a quasi-monolithic cavity.