Robust low-latency data transmission is a challenging problem. The low-latency requirement means that limited source information is available for compression and short channel codes need to be used. In this paper, we study low-latency speech transmission over analog Gaussian wireless channels and propose a deep joint source-channel coding (JSCC) system. The proposed method includes a deep neural network (DNN) that consists of a source-channel encoder which transmits analog coded information over Gaussian wireless channels and a source-channel decoder which recovers data from the received noisy, compressed data. The total source-channel encoding and decoding latency is configurable with respect to the input speech signal length. Simulation studies demonstrate that in low-latency and noisy channel regimes, our proposed JSCC system provides significant gains over state-of-the-art separate digital source-channel communication systems in terms of estimated speech quality and intelligibility. At higher latencies and better channel conditions, the separate coding schemes are better.