We present the construction of a multivariate normalized B-spline basis for the quadratic C¹ continuous spline space defined over a triangulation in ℝˢ (s ≥ 1) with a generalized Powell-Sabin refinement. The basis functions have a local support, they are nonnegative, and they form a partition of unity. The construction can be interpreted geometrically as the determination of a set of s-simplices that must contain a specific set of points. We also propose a family of quasi-interpolants based on this multivariate Powell-Sabin B-spline representation. Their spline coefficients only depend on a set of local function values. The multivariate quasi-interpolants reproduce quadratic polynomials and have an optimal approximation order.