We present exact results on the dynamics of a biased, by an external force F, intruder (BI) in a two-dimensional lattice gas of unbiased, randomly moving hard-core particles. Going beyond the usual analysis of the force-velocity relation, we study the probability distribution P(R_{n}) of the BI displacement R_{n} at time n. We show that despite the fact that the BI drives the gas to a nonequilibrium steady state, P(R_{n}) converges to a Gaussian distribution as n→∞. We find that the variance σ_{x}^{2} of P(R_{n}) along F exhibits a weakly superdiffusive growth σ_{x}^{2}∼ν_{1}nln(n), and a usual diffusive growth, σ_{y}^{2}∼ν_{2}n, in the perpendicular direction. We determine ν_{1} and ν_{2} exactly for arbitrary bias, in the lowest order in the density of vacancies, and show that ν_{1}∼|F|^{2} for small bias, which signifies that superdiffusive behavior emerges beyond the linear-response approximation. We also present analytical arguments predicting a striking field-induced superdiffusive behavior σ_{x}^{2}∼n^{3/2} for two-dimensional stripes and three-dimensional capillaries, which is confirmed by Monte Carlo simulations.