  We consider a perturbed relativistic Kepler problem \begin{equation*}
\dfrac{\mathrm{d}}{\mathrm{d}t}\left(\dfrac{m\dot{x}}{\sqrt{1-|\dot{x}|^2/c^2}}\right)=-\alpha\,
\dfrac{x}{|x|^3}+\varepsilon \, \nabla_x U(t,x), \qquad x \in \mathbb{R}^2
\setminus \{0\}, \end{equation*} where $m, \alpha > 0$, $c$ is the speed of
light and $U(t,x)$ is a function $T$-periodic in the first variable. For
$\varepsilon > 0$ sufficiently small, we prove the existence of $T$-periodic
solutions with prescribed winding number, bifurcating from invariant tori of
the unperturbed problem.
