Diffusion kurtosis imaging (DKI) has undisputed advantages over the more classical diffusion magnetic resonance imaging (dMRI) as witnessed by the fast-increasing number of clinical applications and software packages widely adopted in brain imaging. However, in the neonatal setting, DKI is still largely underutilized, in particular in spinal cord (SC) imaging, because of its inherently demanding technological requirements. Due to its extreme sensitivity to non-Gaussian diffusion, DKI proves particularly suitable for detecting complex, subtle, fast microstructural changes occurring in this area at this early and critical stage of development, which are not identifiable with only DTI. Given the multiplicity of congenital anomalies of the spinal canal, their crucial effect on later developmental outcome, and the close interconnection between the SC region and the brain above, managing to apply such a method to the neonatal cohort becomes of utmost importance. This study will (i) mention current methodological challenges associated with the application of advanced dMRI methods, like DKI, in early infancy, (ii) illustrate the first semi-automated pipeline built on Spinal Cord Toolbox for handling the DKI data of neonatal SC, from acquisition setting to estimation of diffusion measures, through accurate adjustment of processing algorithms customized for adult SC, and (iii) present results of its application in a pilot clinical case study. With the proposed pipeline, we preliminarily show that DKI is more sensitive than DTI-related measures to alterations caused by brain white matter injuries in the underlying cervical SC.