Device-Independent (DI) security is the gold standard of quantum cryptography, providing information-theoretic security based on the very laws of nature. In its highest form, security is guaranteed against adversaries limited only by the no-superluminal signalling rule of relativity. The task of randomness amplification, to generate secure fully uniform bits starting from weakly random seeds, is of both cryptographic and foundational interest, being important for the generation of cryptographically secure random numbers as well as bringing deep connections to the existence of free-will. DI no-signalling proof protocols for this fundamental task have thus far relied on esoteric proofs of non-locality termed pseudo-telepathy games, complicated multi-party setups or high-dimensional quantum systems, and have remained out of reach of experimental implementation. In this paper, we construct the first practically relevant no-signalling proof DI protocols for randomness amplification based on the simplest proofs of Bell non-locality and illustrate them with an experimental implementation in a quantum optical setup using polarised photons. Technically, we relate the problem to the vast field of Hardy paradoxes, without which it would be impossible to achieve amplification of arbitrarily weak sources in the simplest Bell non-locality scenario consisting of two parties choosing between two binary inputs. Furthermore, we identify a deep connection between proofs of the celebrated Kochen-Specker theorem and Hardy paradoxes that enables us to construct Hardy paradoxes with the non-zero probability taking any value in (0,1]. Our methods enable us, under the fair-sampling assumption of the experiment, to realize up to 25 bits of randomness in 20 hours of experimental data collection from an initial private source of randomness 0.1 away from uniform.