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An implicit marching-on-in-time (MOT) algorithm for solving a time domain electric field integral equation pertinent to the analysis of transient scattering and radiation from perfect electrically conducting bodies that are excited by very low frequency signals is presented. Spatial variations in electric current densities are described by loop-tree basis functions to ensure stability of the time marching scheme when geometric discretisation lengths are much smaller than a light time step. Such basis functions were previously employed within explicit MOT solvers. However, to our knowledge, this is the first study to demonstrate their usefulness in tackling time domain integral equation low-frequency breakdown and related stability problems. This is achieved by deploying loop-tree basis functions within an implicit MOT scheme. Numerical examples that demonstrate the proposed scheme's effectiveness in analysing electromagnetically small or mixed scale structures and its ability to operate with time steps determined by the Nyquist criterion as opposed to geometric discretisation rules are presented.