We propose a six-dimensional Universal Extra Dimensions (UED) model compactified on a real projective plane RP2, a two-sphere with its antipodal points being identified. We utilize the Randjbar-Daemi–Salam–Strathdee spontaneous sphere compactification with a monopole configuration of an extra U(1)X gauge field that leads to a spontaneous radius stabilization. Unlike the sphere and the S2/Z2 orbifold compactifications, the massless U(1)X gauge boson is safely projected out. We show how a compactification on a non-orientable manifold results in a chiral four-dimensional gauge theory by utilizing 6D chiral gauge and Yukawa interactions. The resultant Kaluza–Klein mass spectra are distinct from the ordinary UED models compactified on torus. We briefly comment on the anomaly cancellation and also on a possible dark matter candidate in our model.