This paper contains experimental results of in vitro, uniaxial tension of swine brain tissue in finite deformation as well as proposes a new hyper-viscoelastic constitutive model for the brain tissue. The experimental results obtained for two loading velocities, corresponding to strain rates of 0.64 and 0.64×10⁻² s⁻¹, are presented. We believe that these are the first ever experiments of this kind. The applied strain rates were similar to those applied in our previous study, focused on explaining brain tissue properties in compression. The stress−strain curves are convex downward for all extension rates. The tissue response stiffened as the loading speed increased, indicating a strong stress−strain rate dependence. Swine brain tissue was found to be considerably softer in extension than in compression. Previously proposed in the literature brain tissue constitutive models, developed based on experimental data collected in compression are shown to be inadequate to explain tissue behaviour in tension. A new, non-linear, viscoelastic model based on the generalisation of the Ogden strain energy hyperelastic constitutive equation is proposed. The new model accounts well for brain tissue deformation behaviour in both tension and compression (natural strain ∈〈−0.3,0.2〉) for strain rates ranging over five orders of magnitude.