We consider a stochastic partial differential equation (SPDE) model for chemorepulsion, with non-linear sensitivity on the one-dimensional torus. We show that for any suitable initial data there exists a pathwise unique, global solution to the SPDE. Furthermore we show that the associated semi-group is Markov and possesses a unique invariant measure, supported on a Hölder-Besov space of positive regularity, which the solution law converges to exponentially fast. We also establish tail bounds on the invariant measure that are heavier than Gaussian when measured using any $L^p$ norm.