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Anisotropic bulk Dirac (or Weyl) cones in three-dimensional systems have recently gained intense research interest as they are examples of materials with tilted Dirac (or Weyl) cones indicating the violation of Lorentz invariance. In contrast, the studies on anisotropic surface Dirac cones in topological materials which contribute to anisotropic carrier mobility have been limited. By employing angle-resolved photoemission spectroscopy and first-principles calculations, we reveal the anisotropic surface Dirac dispersion in a tetradymite material Ti2Te2P on the (001) plane of the Brillouin zone. We observe quasielliptical Fermi pockets at the (M) over bar point of the Brillouin zone forming the anisotropic surface Dirac cones. Our calculations of the Z(2) indices confirm that the system is topologically nontrivial with multiple topological phases in the same material. In addition, the observed nodal-line-like feature formed by bulk bands makes this system topologically rich.