One of the key challenges of real-time detection and parameter estimation of gravitational waves from compact binary mergers is the computational cost of conventional matched-filtering and Bayesian inference approaches. In particular, the application of these methods to the full signal parameter space available to the gravitational-wave detectors, and/or real-time parameter estimation is computationally prohibitive. On the other hand, rapid detection and inference are critical for prompt follow-up of the electromagnetic and astro-particle counterparts accompanying important transients, such as binary neutron-star and black-hole neutron-star mergers. Training deep neural networks to identify specific signals and learn a computationally efficient representation of the mapping between gravitational-wave signals and their parameters allows both detection and inference to be done quickly and reliably, with high sensitivity and accuracy. In this work we apply a deep-learning approach to rapidly identify and characterize transient gravitational-wave signals from binary neutron-star mergers in real LIGO data. We show for the first time that artificial neural networks can promptly detect and characterize binary neutron star gravitational-wave signals in real LIGO data, and distinguish them from noise and signals from coalescing black-hole binaries. We illustrate this key result by demonstrating that our deep-learning framework classifies correctly all gravitational-wave events from the Gravitational-Wave Transient Catalog, GWTC-1 [Phys. Rev. X 9 (2019), 031040]. These results emphasize the importance of using realistic gravitational-wave detector data in machine learning approaches, and represent a step towards achieving real-time detection and inference of gravitational waves.