I am wondering if the following statements are correct:
(1) zero-order B-splines interpolation is equivalent to nearest-neighbor interpolation. $C^0$ continuity thus is not differentiable.
(2) first-order B-splines interpolation is equivalent to linear interpolation. $C^1$ continuity.
In B-spline terminology, order means degree + 1. Therefore, a degree 3 B-spline has order 4. I am not sure what you meant by first-order B-spline exactly, but I am going to assume you are talking about degree 1 B-spline (as B-spline of order 1 means degree 0, which does not make too much sense). Degree 1 B-spline interpolation is indeed the same as linear interpolation. But it only has $C^0$ continuity.