This paper proposes a distributed Kalman filter (DKF) with enhanced robustness against Byzantine adversaries. A Byzantine agent is a legitimate network agent that, unlike an honest agent, manipulates information before sharing it with neighbors to impair the overall system performance. In contrast to the literature, the DKF is modeled as a distributed optimization problem where resiliency against Byzantine agents is accomplished by employing a total variation (TV) penalty term. We utilize a distributed subgradient algorithm to compute the state estimate and error covariance matrix updates of the DKF. Additionally, we prove that the proposed suboptimal solution converges to a neighborhood of the optimal centralized solution of the Kalman filter (KF) with a bounded radius when Byzantine agents are present. Numerical simulations corroborate the theoretical findings and demonstrate the robustness of the proposed DKF against Byzantine attacks.