This would be the case with the hypothetical negative mass, but there is no evidence such a thing exists in reality. But nevertheless, mathematically a negative $m$ is allowed in general relativity, see the reference Michigan Tech: Negative Mass video and presentation, so one can still play it through as a thought experiment.
Such a hypothetical negative mass would have a repelling gravity, so theoretically with a pair of positive and negative mass you could constantly accelerate both objects in the direction of the positive mass (because the positive mass would be repelled by the negative mass, while the negative mass would be attracted by the positive mass due to the equivalence principle).
The sum of the kinetic energies, negative and postive, of the system would always cancel out to zero, although almost travelling with the speed of light (the technical term for such an accelerating constellation of positive and negative mass objects is runaway pair).
Another weird thing is that if the negative $m$ particle would hit you, it would push you in the direction it came from, since due to the negative kinetic energy the momentum would be in the opposite side of motion.