In many cosmological inference problems, the likelihood (the probability of the observed data as a function of the unknown parameters) is unknown or intractable. This necessitates approximations and assumptions, which can lead to incorrect inference of cosmological parameters, including the nature of dark matter and dark energy, or create artificial model tensions. Likelihood-free inference covers a novel family of methods to rigorously estimate posterior distributions of parameters using forward modelling of mock data. We present likelihood-free cosmological parameter inference using weak lensing maps from the Dark Energy Survey (DES) Science Verification data, using neural data compression of weak lensing map summary statistics. We explore combinations of the power spectra, peak counts, and neural compressed summaries of the lensing mass map using deep convolution neural networks. We demonstrate methods to validate the inference process, for both the data modelling and the probability density estimation steps. Likelihood-free inference provides a robust and scalable alternative for rigorous large-scale cosmological inference with galaxy survey data (for DES, Euclid, and LSST). We have made our simulated lensing maps publicly available.