= Solution
For constant laboratory <force> $\mathbf F$ and initial rest, <integration> gives <three-momentum> $\mathbf p=\mathbf Ft$. The <energy–momentum relation> gives $E=\sqrt{m^2c^4+c^2|\mathbf F|^2t^2}$, and $\mathbf v=c^2\mathbf p/E$, so
$$
\boxed{\mathbf v(t)=\frac{\mathbf Ft/m}{\sqrt{1+|\mathbf F|^2t^2/(m^2c^2)}}.}
$$
For $\mathbf F\ne0$ and $t\ge0$, its <speed> increases monotonically, is strictly below the <speed of light> at every finite time, and obeys $\lim_{t\to\infty}|\mathbf v(t)|=c$. Thus $c$ is the limiting supremum, rather than a <speed> attained at a finite time. The degenerate case $\mathbf F=0$ leaves the particle at rest and does not have this limit.
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