Range-based SLAM is a well-established technique for estimating the pose of a mobile robot operating in an unknown environment. Current state-of-the-art solutions use occupancy-grid maps to represent the world. While fast and accurate, their performance is limited by two facts. First, in an occupancy-grid map measurements have to be discretised into cell resolution. Second, online pose estimation, which relies on scan-to-map alignment, typically requires smoothing/interpolating the discrete grid-map. This letter presents a SLAM technique that builds on top of a B-spline surface map. The local properties of splines and the inherent smoothness of their basis function handle the aforementioned problems, without significant increase in the computational cost. Through qualitative and quantitative tests using public data sets we show that the proposed B-spline SLAM is an affordable technique that delivers accurate results at sensor rate speed.