We show that the use of higher dimensional wrapped branes can significantly extend the inflaton field range compared to brane inflation models which use D3-branes. We construct a simple inflationary model in terms of 5-branes wrapping a 2-cycle and traveling towards the tip of the Klebanov-Strassler throat. Inflation ends when the branes reach the tip of the cone and self-annihilate. Assuming a quadratic potential for the brane it is possible to match the cosmic microwave background data in the Dirac-Born-Infeld regime, but we argue that the backreaction of the brane is important and cannot be neglected. This scenario predicts a strong non-Gaussian signal and possibly detectable gravitational waves.
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