We introduce a new Markov chain Monte Carlo (MCMC) sampler for infinite-dimensional inverse problems. Our new sampler is based on the affine invariant ensemble sampler, which uses interacting walkers to adapt to the covariance structure of the target distribution. We extend this ensemble sampler for the first time to infinite-dimensional function spaces, yielding a highly efficient gradient-free MCMC algorithm. Because our new ensemble sampler does not require gradients or posterior covariance estimates, it is simple to implement and broadly applicable.