Following the computation revolution in machine learning, the delayed feedback reservoir (DFR) computing system has shown its promising perspectives toward mimicking our mammalian brains, with comparable performance to other traditional neuromorphic computing systems. The DFR, which is embedded with a single Mackey-Glass (MG) nonlinear function with dynamic delay feedback loop, does not only have resemblance to mammalian brains, but also exhibits the near chaotic regime behavior. Inspired by the traditional DFR computing system and MG nonlinear function, in this work, we proposed a spiking DFR (S-DFR) computing system that built on the standard Global Foundries 130 nm CMOS technology. Our S-DFR computing system adopts a temporal coding scheme, where the pre- and post-neuron signals are represented by the digitized pulse train with alterable time intervals. The proposed S-DFR demonstrates its high energy efficiency with less design area, compared to the state-of-the-art MG based DFR and Liquid State Machine (LSM) hardware implementation; most importantly, the experimental results illustrate the high nonlinearity and the delay property of the proposed S-DFR.