We present the first modelocked backside-cooled GaSb VECSEL (Vertical External Cavity Surface Emitting Laser) operating above <inline-formula> <tex-math notation="LaTeX">$2 ~\mu \text{m}$ </tex-math></inline-formula>. Using a two quantum well InGaSb SESAM (SEmiconductor Saturable Absorber Mirror) in a V-shaped cavity arrangement we obtain femtosecond modelocking at a center wavelength of 2061 nm, with pulses as short as 324 fs, an average output power as high as 65 mW at a repetition rate of 3 GHz. An operation in a picosecond regime is further demonstrated with an even higher average output power of up to 260 mW and 4.5-ps pulses. We perform a characterization in both the femto- and picosecond regimes with measurements of the pulse duration, the optical spectrum, and the pulse repetition rate (frequency and time domain analysis) at low and high power. In this context, the influence of intracavity group delay dispersion (GDD) is investigated. We find that GDD has a strong influence on the pulse duration (fs or ps regime) and thus on the performance of our SESAM-modelocked 2-<inline-formula> <tex-math notation="LaTeX">$\mu \text{m}$ </tex-math></inline-formula> GaSb VECSEL.