= Solution
With $\mathbf B=0$ and $\mathbf E=(0,0,E)$, the spatial <Lorentz force> equation gives
$$
p_x=p_0,\qquad p_y=0,\qquad p_z=qEt.
$$
The <relativistic energy-momentum relation> therefore yields
$$
\boxed{
\mathcal E(t)=
\sqrt{\mathcal E_0^2+c^2q^2E^2t^2}
},
\qquad
\mathcal E_0=\sqrt{m^2c^4+c^2p_0^2}.
$$
Since $\mathbf v=c^2\mathbf p/\mathcal E$,
$$
\dot z=\frac{c^2qEt}{\mathcal E(t)}.
$$
Integration from the initial <position> $z(0)=0$ gives
$$
\boxed{
z(t)=
\frac{\sqrt{\mathcal E_0^2+c^2q^2E^2t^2}-\mathcal E_0}{qE}
}.
$$
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