Low space computation and branching program
Denoting the class of functions computed by branching programs of size $f(n)$ by $\text{BP}(f)$,
the best known bound seems to be the trivial $\text{DSPACE}(S(n)) \subseteq \text{BP}(2^{O(S(n))})$.
This matches the known inclusions $\text{DSPACE}(\log n) = \text{L} \subseteq \text{L}/poly = \text{BP}(poly)$; if a better bound were known then I would expect it to have been noted by Razborov in his FCT 1991 survey (preprint), or in Jukna's 2012 BFC textbook.