We show that, if \(\alpha > 0\) is a real number, \(n \ge 2\) and \(\ell \ge 2\) are integers, and \(q\) is a prime power, then every simple matroid \(M\) of sufficiently large rank, with no \(U_{2,\ell}\)-minor, no rank-\(n\) projective geometry minor over a larger field than \(\GF(q)\), and satisfying \(|M| \ge \alpha q^{r(M)}\), has a rank-\(n\) affine geometry restriction over \(\GF(q)\). This result can be viewed as an analogue of the Multidimensional Density Hales-Jewett Theorem for matroids.