Josephson junction arrays constitute a promising platform to explore a large class of physical phenomena, from quantum phase transitions to non-linear quantum optics in the microwave domain. We will report on the measurement of a fully-tunable model system where a long chain of several thousands linear Josephson elements, acting as a strongly dissipative environment, is terminated by a small Josephson junction endowed with a strong non-linearity, acting as a single impurity. From microwave spectroscopic measurements, we extract the phase shift and the inelastic losses induced by the impurity onto the linear modes of the array. In agreement with a microscopic modeling of the circuit, we put into evidence a huge renormalization of the Josephson tunnel energy at the impurity site, and show that the associated enhancement of phase fluctuations provides the dominant dissipation mechanism in the array.