We use a new (to our knowledge) fabrication method of a single-walled carbon nanotube (SWCNT) absorber without polymer to sustain high-power illumination. Using a series of saturable absorbers (SAs) incorporating different amounts of SWCNTs, we demonstrate mode-locking for a $\mathrm{Nd}\text{:}{\mathrm{GdVO}}_4$ laser in the $1\mathrm{\mu m}$ spectral range. Continuous- wave mode-locking (CWML) pulses with a maximum output power of $3.6\mathrm{W}$ at $1063\mathrm{nm}$ and high noise extinction of $61\mathrm{dB}$ has been achieved to give the highest pulse peak power of $3.6\mathrm{kW}$ and pulse energy of $30\mathrm{nJ}$ under $15\mathrm{W}$ pumping. To our knowledge, this is the highest CWML output power with SWCNT-SAs reported. The measured nonlinear absorption of the SWCNT-SAs shows a modulation depth of $\sim 3\%$ with subpicosecond recovery time.