X-ray fluorescence computed tomography (XFCT) is a high sensitivity imaging modality for high-atomic elements such as gadolinium (Gd) or gold (Au). In order to improve the contrast of x-ray fluorescent (XRF) signal in the raw projection data, common XFCT systems use the single-pixel x-ray spectrometer to record XRF photons stimulated by the pencil-beam x-ray source and achieve line-by-line scan of the whole object by translating the x-ray source or the object. However, this kind of design results in waiting time of translation device. One improvement would be to replace the traditional x-ray tube with the spatially distributed multi-beam x-ray source and scan the whole object by switching the exposure of each focus. In this study, we present a design of XFCT system using the spatially distributed multi-beam source and a XFCT imaging experiment was performed to investigate its feasibility. Each cold cathode of the x-ray source used in this study was made of carbon nanotubes and each focus was independently controlled by the electronic control system. The incident beam was collimated by a pinhole array to produce pencil-beam source. The object scanned by the system was a PMMA cylinder (8cm in diameter) with Gd (20mg/mL) and I (100mg/mL) insertions. Results show that the distribution of I and Gd in a PMMA phantom was successfully reconstructed but the imaging performance was limited by the collimation and exposure mode of current distributed x-ray source. The practicality of current distributed x-ray source used for XFCT scan and further optimization of the proposed system are discussed according to the experimental results.