We study the out-of-equilibrium dynamics of an impurity driven by a constant external force through a system of homogeneous, weakly-interacting bosons in one spatial dimension. The impurity-boson interaction is assumed to be attractive. We show that the impurity exhibits drifted Bloch oscillations over a wide range of forces in the absence of a lattice. We explain the mechanism underlying the Bloch oscillations, which occur via the formation of a depletion region attached to the density peak centered at the impurity. We analyze the behavior of the drift velocity, the Bloch amplitude, and the time period of oscillations in a wide range of forces and other system parameters. In contrast to the case of repulsive impurity-boson interaction, the drift velocity exhibits a sub-linear dependence on a weak applied force, $V_d\sim F^{\alpha }$ with a positive exponent $\alpha$ smaller than unity. The drift velocity monotonically increases with force, though the scaling behavior varies considerably across different regimes of $F$. Moreover, the amplitude of the velocity oscillations displays rich behavior: it first undergoes a decay with force, reaches a minimum, and then presents a revival, increasing with force. The velocity oscillations of an impurity exceeding the critical mass differ significantly in shape and drift velocity behavior as a function of force.