Kernel phase interferometry (KPI) is a data processing technique that allows for the detection of asymmetries (such as companions or disks) in high-Strehl images, close to and within the classical diffraction limit. We show that KPI can successfully be applied to hyperspectral image cubes generated from integral field spectrographs (IFSs). We demonstrate this technique of spectrally dispersed kernel phase by recovering a known binary with the SCExAO/CHARIS IFS in high-resolution K-band mode. We also explore a spectral differential imaging (SDI) calibration strategy that takes advantage of the information available in images from multiple wavelength bins. Such calibrations have the potential to mitigate high-order, residual systematic kernel phase errors, which currently limit the achievable contrast of KPI. The SDI calibration presented is applicable to searches for line emission or sharp absorption features and is a promising avenue toward achieving photon-noise-limited kernel phase observations. The high angular resolution and spectral coverage provided by dispersed kernel phase offers opportunities for science observations that would have been challenging to achieve otherwise.