We present a wavelength-tunable, external-cavity GaSb-based quantum-well laser operating near 3.2 μm. The laser setup consists of an intra-cavity grating in Littman-Metcalf configuration and a cascade pumped GaSb-based gain chip with a narrow-ridge waveguide. The narrow-ridge waveguide has a length of 2 mm and width of 7.5 μm. Cascade pumping is realized with three type-I quantum-wells, using one quantum-well per cascade stage. The laser provides continuous-wave output powers up to 8 mW and slope-efficiencies of 13 % at room temperature. Laser operation is demonstrated over a wavelength range of more than 300 nm, using continuous-wave and pulsed operation regimes.