No, it is not split (except in a few degenerate cases like $g=0$; let's assume that $g \geq 2$ for the moment just to be careful). It is a central extension, so if it was split then the abelianization of the pure mapping class group of $S_{g,b}$ would contain a copy of $\mathbb{Z}^b$; however, this abelianization is trivial

No, it is not split (except in a few degenerate cases like $(g,n) = (0,1)$ or $(g,n)=(0,2)$; let's assume that $g \geq 2$ for the moment just to be careful). It is a central extension, so if it was split then the abelianization of the pure mapping class group of $S_{g,b}$ would contain a copy of $\mathbb{Z}^b$; however, this abelianization is trivial.