The study of noise sensitivity of Boolean functions was initiated in a seminal paper of Benjamini, Kalai and Schramm, published in 1999. While this study has revealed fascinating phenomena in the context of Bernoulli percolation, few results have been obtained regarding other random spatial processes. In this paper we prove the first instance of noise sensitivity for a spatial growth process associated to the Kardar-Parisi-Zhang class of universality. More specifically, we show that travel times in geometric last-passage percolation are noise sensitive with respect to a perturbation acting on a Bernoulli encoding of the geometric weights. Our method of proof includes a generalisation of the celebrated Benjamini-Kalai-Schramm noise sensitivity/influence theorem, and precise bounds on the probability of a given vertex being on a geodesic, which we believe to be of independent interest.