  In a superconductor that lacks inversion symmetry, the spatial part of the
Cooper pair wave function has a reduced symmetry, allowing for the mixing of
spin-singlet and spin-triplet Cooper pairing channels and thus providing a
pathway to a non-trivial superconducting state. Materials with a
non-centrosymmetric crystal structure and with strong spin-orbit coupling are a
platform to realize these possibilities. Here, we report the synthesis and
characterisation of high quality crystals of Sn$_4$As$_3$, with
non-centrosymmetric unit cell ($R3m$). We have characterised the normal and
superconducting state using a range of methods. Angle-resolved photoemission
spectroscopy shows a multiband Fermi surface and the presence of two surface
states, confirmed by Density-functional theory calculations. Specific heat
measurements reveal a superconducting critical temperature of $T_c\sim 1.14$ K
and an upper critical magnetic field of $H_c\gtrsim 7$ mT, which are both
confirmed by ultra-low temperature scanning tunneling microscopy and
spectroscopy. Scanning tunneling spectroscopy shows a fully formed
superconducting gap, consistent with conventional $s$-wave superconductivity.
