  The Maker-Breaker domination game is played on a graph $G$ by two players,
called Dominator and Staller, who alternately choose a vertex that has not been
played so far. Dominator wins the game if his moves form a dominating set.
Staller wins if she plays all vertices from a closed neighborhood of a vertex
$v \in V(G)$. Dominator's fast winning strategies were studied earlier. In this
work, we concentrate on the cases when Staller has a winning strategy in the
game. We introduce the invariant $\gamma'_{\rm SMB}(G)$ (resp., $\gamma_{\rm
SMB}(G)$) which is the smallest integer $k$ such that, under any strategy of
Dominator, Staller can win the game by playing at most $k$ vertices, if Staller
(resp., Dominator) plays first on the graph $G$.
  We prove some basic properties of $\gamma_{\rm SMB}(G)$ and $\gamma'_{\rm
SMB}(G)$ and study the parameters' changes under some operators as taking the
disjoint union of graphs or deleting a cut vertex. We show that the inequality
$\delta(G)+1 \le \gamma'_{\rm SMB}(G) \le \gamma_{\rm SMB}(G)$ always holds and
that for every three integers $r,s,t$ with $2\le r\le s\le t$, there exists a
graph $G$ such that $\delta(G)+1 = r$, $\gamma'_{\rm SMB}(G) = s$, and
$\gamma_{\rm SMB}(G) = t$. We prove exact formulas for $\gamma'_{\rm SMB}(G)$
where $G$ is a path, or it is a tadpole graph which is obtained from the
disjoint union of a cycle and a path by adding one edge between them.
