Innovation is the ability to use a new behaviour, or use an existing behaviour in a new context. Innovation, as an aspect of behavioural flexibility, could be important for allowing animals to cope with rapid environmental changes. Surprisingly, few studies have focused on how innovation ability is affected by task complexity. We investigated innovation ability across multiple tasks of varying complexity in a native Australian rodent, the fawn-footed mosaic-tailed rat (Melomys cervinipes). We predicted that mosaic-tailed rats would be capable of innovating because they live in complex habitats and can exploit disturbed and changing environments. However, we also predicted that the success rate of innovating would decrease as task complexity increased. Mosaic-tailed rats were exposed to six novel problems: cylinder, matchbox, obstruction test, pillar, tile and lever (the last three presented in a Trixie dog activity board), which represented increasing complexity. We counted the number of individuals that could solve at least one task, compared individuals for solving efficiency and latency to solve, and compared the solving success of each task. All mosaic-tailed rats could innovate. However, solving success differed between individuals, with some solving every task and others only solving one. Solving success rate was significantly higher in the simplest task (pillar) compared to the most complicated task (lever). There was no effect of sex or sampling condition on innovation. This study is the first to demonstrate innovation ability across task complexity in an Australian rodent and provides promising avenues for future studies of innovation.