A Weyl semimetal is a topologically non-trivial phase of matter that hosts mass-less Weyl fermions, the particles that remained elusive for more than 80 years since their theoretical discovery. The Weyl semimetals exhibit unique transport and magneto-transport properties and remarkably high surface spin polarization. Here we show that a unique mesoscopic superconducting phase with a critical temperature up to 7 K can be realized by forming metallic point contacts with silver (Ag) on single crystals of TaAs, while neither Ag nor TaAs are superconductors. The Andreev reflection spectra obtained from such point contacts are fitted well within a modified Blonder-Tinkham-Klapwijk (BTK) model with a superconducting energy gap up to 1.2 meV. The analysis within this model also reveals high transport spin polarization up to 60% indicating a spin polarized supercurrent flowing through the point contacts on TaAs. Such point contacts also show a large anisotropic magnetoresistance (AMR) originating from the spin polarized current. Therefore, apart from the discovery of a novel mesoscopic superconducting phase and its coexistence with a large spin polarization, our results also show that the point contacts on Weyl semimetals are potentially important for applications in spintronics.