Neural likelihood estimation methods for simulation-based inference can suffer from performance degradation when the modeled data is very high-dimensional or lies along a lower-dimensional manifold, which is due to the inability of the density estimator to accurately estimate a density function. We present Surjective Sequential Neural Likelihood (SSNL) estimation, a novel member in the family of methods for simulation-based inference (SBI). SSNL fits a dimensionality-reducing surjective normalizing flow model and uses it as a surrogate likelihood function, which allows for computational inference via Markov chain Monte Carlo or variational Bayes methods. Among other benefits, SSNL avoids the requirement to manually craft summary statistics for inference of high-dimensional data sets, since the lower-dimensional representation is computed simultaneously with learning the likelihood and without additional computational overhead. We evaluate SSNL on a wide variety of experiments, including two challenging real-world examples from the astrophysics and neuroscience literatures, and show that it either outperforms or is on par with state-of-the-art methods, making it an excellent off-the-shelf estimator for SBI for high-dimensional data sets.