Easy to construct and optimally convergent generalisations of B-splines to unstructured meshes are essential for the application of isogeometric analysis to domains with non-trivial topologies. Nonetheless, especially for hexahedral meshes, the construction of smooth and optimally convergent isogeometric analysis basis functions is still an open question. We introduce a simple partition of unity construction that yields smooth blended B-splines, referred to as SB-splines, on semi-structured quadrilateral and hexahedral meshes, i.e. on mostly structured meshes with sufficiently separated unstructured regions. To this end, we first define the mixed smoothness B-splines that are C⁰ continuous in the unstructured regions of the mesh but have higher smoothness everywhere else. Subsequently, the SB-splines are obtained by smoothly blending in the physical space the mixed smoothness B-splines with Bernstein bases of equal degree. One of the key novelties of our approach is that the required smooth weight functions are assembled from the available smooth B-splines on the unstructured mesh. The SB-splines are globally smooth, non-negative, have no breakpoints within the elements and reduce to conventional B-splines away from the unstructured regions of the mesh. Although we consider only quadratic mixed smoothness B-splines in this paper, the construction generalises to arbitrary degrees. We demonstrate the excellent performance of SB-splines studying Poisson and biharmonic problems on semi-structured quadrilateral and hexahedral meshes, and numerically establishing their optimal convergence in one and two dimensions.