The two-dimensional kagome lattice is an experimental playground for novel physical phenomena, from frustrated magnetism and topological matter to chiral charge order and unconventional superconductivity. A newly identified kagome superconductor, Ta₂V_3.1Si_0.9 has recently gained attention for possessing a record high critical temperature, T_C = 7.5 K for kagome metals at ambient pressure. In this study we conducted a series of muon spin rotation measurements to delve deeper into understanding the superconducting and normal state properties of Ta₂V_3.1Si_0.9. We demonstrate that Ta₂V_3.1Si_0.9 is a bulk superconductor with either a s+s-wave or anisotropic s-wave gap symmetry, and has an unusual paramagnetic shift in response to external magnetic fields in the superconducting state. Additionally, we observe an exceptionally low superfluid density − a distinctive characteristic of unconventional superconductivity − which remarkably is comparable to the superfluid density found in hole-doped cuprates. In its normal state, Ta₂V_3.1Si_0.9 exhibits a significant increase in the zero-field muon spin depolarisation rate, starting at approximately 150 K, which has been observed in other kagome-lattice superconductors, and therefore hints at possible hidden magnetism. These findings characterise Ta₂V_3.1Si_0.9 as an unconventional superconductor and a noteworthy new member of the vanadium-based kagome material family.