The -defective -painting game on a graph is played by two players: Lister and Painter. Initially, each vertex has tokens and is uncoloured. In each round, Lister chooses a set of uncoloured vertices and removes one token from each chosen vertex. Painter colours a subset of which induces a subgraph of maximum degree at most . Lister wins the game if at the end of some round, a vertex has no more tokens left, and is uncoloured. Otherwise, at some round, all vertices are coloured and Painter wins. We say is -defective -paintable if Painter has a winning strategy in this game. This paper proves that for each surface , there is a constant such that graphs embedded in with edge-width at least are -defective -paintable.