Separating disks in 3-manifolds

Question

Let M be a (smooth or PL) connected three manifold with boundary, such that one boundary component is a sphere S. Let D be a properly embedded disk whose boundary lies on S. Must D separate M into distinct connected components? Intuitively the answer seems clear, but I can't figure out a proof

Answer 1

Consider $M=S^2\times S^1-B^3$, which is a connected $3$-manifold with boundary $S^2$. One of the $S^2\times x$ slices runs through the middle of the $B^3$, and the complement of this slice in $M$ is a properly embedded disk that does not separate $M$.