Advances in materials science and memory devices work in tandem for the evolution of Artificial Intelligence systems. Energy-efficient computation is the ultimate goal of emerging memristor technology, in which the storage and computation can be done in the same memory crossbar. In this work, an analog memristor device is fabricated utilizing the unique characteristics of single-wall carbon nanotubes (SWCNTs) to act as the switching medium of the device. Via the planar structure, the memristor device exhibits analog switching ability with high state stability. The device’s conductance and capacitance can be tuned simultaneously, increasing the device's potential and broadening its applications' horizons. The multi-state storage capability and long-term memory are the key factors that make the device a promising candidate for bio-inspired computing applications. As a demonstrator, the fabricated memristor is deployed in spiking neural networks (SNN) to exploit its analog switching feature for energy-efficient classification operation. Results reveal that the computation-in-memory implementation performs Vector Matrix Multiplication with 95% inference accuracy and few femtojoules per spike energy efficiency. The memristor device presented in this work opens new insights towards utilizing the outstanding features of SWCNTs for efficient analog computation in deep learning systems.