I will discuss the existence of hyperkähler structures near the identity section of a holomorphic symplectic groupoid, and show that they always exist in complex dimension four, i.e. when the base is a holomorphic Poisson surface. The hyperkähler structure is obtained by constructing the twistor space by pulling back specific deformations of the Poisson surface adapted from Hitchin's unobstructedness result. In the special case of the zero Poisson structure, we recover the Feix--Kaledin hyperkähler structure on the cotangent bundle of a Kähler manifold.