  In a recent paper of Ellenberg, Oberlin, and Tao, the authors asked whether
there are Besicovitch phenomena in F_q[[t]]^n. In this paper, we answer their
question in the affirmative by explicitly constructing a Kakeya set in
F_q[[t]]^n of measure 0. Furthermore, we prove that any Kakeya set in
F_q[[t]]^2 or Z_p^2 is of Minkowski dimension 2.
