Electrical impedance tomography (EIT) is an imaging method that can be applied over stretchable conductive-fabric materials to realize soft and wearable pressure sensors through current injections and voltage measurements at electrodes placed at the boundary of a conductive medium. In common EIT systems, the voltage data are serially measured by means of multiplexers, and are hence collected at slightly different times, which affects the real-time performance of the system. They also tend to have complicated hardware, which increases power consumption. In this paper, we present our design of a 16-electrode high-speed EIT system that simultaneously implements constant current injection and differential potential measurements. This leads to a faster, simpler-to-implement and less-noisy technique, when compared with traditional EIT approaches. Our system consists of a Howland current pump with two multiplexers for a constant DC current supply, and a data acquisition card. It guarantees a data collection rate of 78 frames/s. The results from our conductive stretchable fabric sensor show that the system successfully performs voltage data collection with a mean signal-to-noise ratio (SNR) of 55 dB, and a mean absolute deviation (MAD) of 0.5 mV. The power consumption can be brought down to 3 mW; therefore, it is suitable for battery-powered applications. Finally, pressure contacts over the sensor are properly reconstructed, thereby validating the efficiency of our EIT system for soft and stretchable sensor applications.