We report on the characteristics of (AlGaIn)(AsSb)-based optically pumped vertical-external-cavity surface-emitting lasers (VECSELs) emitting at wavelengths around 2.35 μm. For barrier-pumped VECSELs mounted substrate-side down without substrate thinning, typical room temperature cw output powers of 2 mW were achieved, limited by premature thermal rollover due to strong device overheating. The thermal impedance of the VECSEL semiconductor chip could be considerably reduced by bonding an intra-cavity polycrystalline CVD diamond heat spreader to the top surface of the chip. This way, at -18°C a maximum cw output power of 0.6 W and a slope efficiency of 10% were obtained for a multiple transverse mode output beam limited by the available pump power rather than by thermal rollover. Optimising the resonator for TEM00 mode operation (M2≈1.1), an output power exceeding 0.4 W was achieved. To reduce the large quantum deficit of more than 50% inherent to barrier-pumped (1.06 μm pump wavelength) GaSb-based VECSELs which emit at wavelengths above 2 μm, we realized a first in-well pumped (AlGaIn)(AsSb) VECSEL where the pump light is absorbed directly in the quantum wells, with the amount of absorbed light enhanced by a higher order microcavity resonance. Using a pump wavelength of 1.94 μm, the quantum deficit is reduced to only 18% and an output power of 5mW, limited by the available pump power, and a slope efficiency of 10% were achieved. Further optimisation of the pump optics is expected to result in a significant increase in device performance.