= Solution
If $e\mathbf E\ne0$, then $\mathbf p(t)=e\mathbf E t+O(1)$, and the <relativistic energy-momentum relation> gives
$$
\boxed{\lim_{t\to\infty}\mathbf v(t)=c\frac{e\mathbf E}{|e\mathbf E|}.}
$$
The particle approaches the <speed of light> in the direction of the electric force, independently of its initial velocity. For positive $e$ the limiting direction is along $\mathbf E$; for negative $e$ it is opposite. A massive particle never reaches that limiting speed in finite time. If the force is zero, the velocity remains $\mathbf v_0$ instead.
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