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Kernel phase imaging (KPI) enables the direct detection of substellar companions and circumstellar dust close to and below the classical (Rayleigh) diffraction limit. We present a kernel phase analysis of JWST NIRISS full pupil images taken during the instrument commissioning and compare the performance to closely related NIRISS aperture masking interferometry (AMI) observations. For this purpose, we develop and make publicly available the custom "Kpi3Pipeline" enabling the extraction of kernel phase observables from JWST images. The extracted observables are saved into a new and versatile kernel phase FITS file (KPFITS) data exchange format. Furthermore, we present our new and publicly available "fouriever" toolkit which can be used to search for companions and derive detection limits from KPI, AMI, and long-baseline interferometry observations while accounting for correlated uncertainties in the model fitting process. Among the four KPI targets that were observed during NIRISS instrument commissioning, we discover a low-contrast (~1:5) close-in (~1 $λ/D$) companion candidate around CPD-66~562 and a new high-contrast (~1:170) detection separated by ~1.5 $λ/D$ from 2MASS~J062802.01-663738.0. The 5-$σ$ companion detection limits around the other two targets reach ~6.5 mag at ~200 mas and ~7 mag at ~400 mas. Comparing these limits to those obtained from the NIRISS AMI commissioning observations, we find that KPI and AMI perform similar in the same amount of observing time. Due to its 5.6 times higher throughput if compared to AMI, KPI is beneficial for observing faint targets and superior to AMI at separations >325 mas. At very small separations (<100 mas) and between ~250-325 mas, AMI slightly outperforms KPI which suffers from increased photon noise from the core and the first Airy ring of the point-spread function.