Consider the following game played on the edges of a lattice $\Lambda$. Two players, Maker and Breaker, alternate turns claiming unclaimed edges of $\Lambda$. Whenever Breaker claims an edge, it is removed from $\Lambda$, while Maker protects edges by claiming them. Breaker wins the game if the connected component of the origin becomes finite at some turn. Maker then wins by guaranteeing that the connected component of the origin remains infinite forever. Who has a winning strategy, say, when $\Lambda = \mathbb{Z}^2$? What if before the game starts, the edges of $\Lambda$ are kept independently with probability $p$ and removed with probability $1 - p$? What about other lattices $\Lambda$? What if Maker can claim $m \geq 1$ edges in her turn and Breaker claims $b \geq 1$ edges in his turn? We consider these and similar problems, provide some answers, and pose many open problems.