In this paper, the elastic deformation behaviour of a recently developed beta-type titanium alloy Ti-24Nb-4Zr-7.9Sn (wt.%) that consists of non-toxic elements and is intended for biomedical applications is described. Tensile tests show that this alloy in the as hot-rolled state exhibits peculiar non-linear elastic behaviour with maximum recoverable strain up to 3.3% and incipient Young's modulus of 42GPa. Solution treatment at high temperature has trivial effect on super-elasticity but decreases strength and slightly increases the incipient Young's modulus. Ageing treatment in the (alpha+beta) two-phase field increases both strength and Young's modulus and results in a combination of high strength and relatively low elastic modulus. In spite of the formation of the alpha phase, short time ageing has no effect on super-elasticity, whereas the non-linear elastic behaviour transforms gradually to normal linear elasticity with the increase of ageing time. We suggest sluggish, partially reversible processes of stress-induced phase transformation and/or incipient kink bands as the origin of the above peculiar elastic behaviour.