Demand for highly energy-efficient coprocessor for the inference computation of deep neural networks is increasing. We propose the time-domain neural network (TDNN), which employs time-domain analog and digital mixed-signal processing (TDAMS) that uses delay time as the analog signal. TDNN not only exploits energy-efficient analog computing, but also enables fully spatially unrolled architecture by the hardware-efficient feature of TDAMS. The proposed fully spatially unrolled architecture reduces energy-hungry data moving for weight and activations, thus contributing to significant improvement of energy efficiency. We also propose useful training techniques that mitigate the non-ideal effect of analog circuits, which enables to simplify the circuits and leads to maximizing the energy efficiency. The proof-of-concept chip shows unprecedentedly high energy efficiency of 48.2 TSop/s/W.