In this paper, we consider the state estimation problem of the systems with time-varying heavy-tailed non-Gaussian noise. The Kalman filtering algorithms based on the maximum correntropy criterion are characterized by their small computational burden and the ability to suppress many types of non-Gaussian noise. However, most of the existing methods are designed using Gaussian kernel functions and the existing kernel bandwidths are limited by its adaptive selection ability. To overcome the problems associated with existing methods, we adopt the student’s $t $ kernel function instead of the Gaussian kernel function commonly used in the existing algorithms to fully capture the information of the heavy-tailed non-Gaussian noise. Based on this, we define a new cost function and derive the student’s $t $ kernel maximum correntropy Kalman filter. Subsequently, to enhance the estimation accuracy of the proposed algorithm under the time-varying heavy-tailed non-Gaussian noise, the interacting multiple model method is employed to select the kernel bandwidths adaptively by running multiple sub-filters with different kernel bandwidths, where the credibilities of all sub-filters are updated using the likelihood functions, and the final posterior estimation is obtained by using the weighted fusion of the sub-filters’ estimation. The target tracking simulation experiments show that the proposed algorithm has better estimation accuracy compared with related algorithms.