We report successful results from using deep learning neural networks (DLNNs) to learn, purely by observation, the behavior of profitable traders in an electronic market closely modelled on the limit-order-book (LOB) market mechanisms that are commonly found in the real-world global financial markets for equities (stocks & shares), currencies, bonds, commodities, and derivatives. Successful real human traders, and advanced automated algorithmic trading systems, learn from experience and adapt over time as market conditions change; our DLNN learns to copy this adaptive trading behavior. A novel aspect of our work is that we do not involve the conventional approach of attempting to predict time-series of prices of tradeable securities. Instead, we collect large volumes of training data by observing only the quotes issued by a successful sales-trader in the market, details of the orders that trader is executing, and the data available on the LOB (as would usually be provided by a centralized exchange) over the period that the trader is active. In this paper we demonstrate that suitably configured DLNNs can learn to replicate the trading behavior of a successful adaptive automated trader, an algorithmic system previously demonstrated to outperform human traders. We also demonstrate that DLNNs can learn to perform better (i.e., more profitably) than the trader that provided the training data. We believe that this is the first ever demonstration that DLNNs can successfully replicate a human-like, or super-human, adaptive trader operating in a realistic emulation of a real-world financial market. Our results can be considered as proof-of-concept that a DLNN could, in principle, observe the actions of a human trader in a real financial market and over time learn to trade equally as well as that human trader, and possibly better.