In 1928, Dirac proposed a wave equation to describe relativistic electrons(1). Shortly afterwards, Klein solved a simple potential step problem for the Dirac equation and encountered an apparent paradox: the potential barrier becomes transparent when its height is larger than the electron energy. For massless particles, backscattering is completely forbidden in Klein tunnelling, leading to perfect transmission through any potential barrier(2,3). The recent advent of condensed-matter systems with Dirac-like excitations, such as graphene and topological insulators, has opened up the possibility of observing Klein tunnelling experimentally(4-6). In the surface states of topological insulators, fermions are bound by spin-momentum locking and are thus immune from backscattering, which is prohibited by time-reversal symmetry. Here we report the observation of perfect Andreev reflection in point-contact spectroscopy-a clear signature of Klein tunnelling and a manifestation of the underlying 'relativistic' physics of a proximity-induced superconducting state in a topological Kondo insulator. Our findings shed light on a previously overlooked aspect of topological superconductivity and can serve as the basis for a unique family of spintronic and superconducting devices, the interface transport phenomena of which are completely governed by their helical topological states.