Topological superconductors showing Majorana fermions or bound states at the edge or surface are promising components for potential quantum computation and thus attract much attention in physical sciences. Recently, topological Weyl fermion states have been discovered in nonsuperconducting TaP crystals, which may offer a great platform to investigate topological nontrivial superconductivity if superconductivity can be induced wherein. Here we report the discovery of nonmagnetic and ferromagnetic tips induced unconventional interface superconductivity around the nanoscale point contact region on TaP crystals. The tip-induced superconductivity (TISC) at the contact is enhanced with the ferromagnetic Ni tip, comparing with nonmagnetic Au and PtIr tips. The point contact spectroscopy shows evidences of anisotropic superconductivity with the mixed spin-triplet pairing, indicating unconventional superconducting nature. Moreover, the Shubnikov--de Haas oscillations in magnetoresistance are detected from the tip-point-contact measurement, which demonstrate that the nonzero Berry phase or topological nontrivial property of TaP maintains under the tip point contact. The theoretical analysis shows that surface superconductivity plays a major role in this TISC and there exists a large spin-triplet mixing in the gap function. Therefore, our work suggests an intriguing Weyl surface superconducting state on TaP.