Answer
The electron is not relativistic.
Work Step by Step
The electrostatic attraction provides the centripetal force keeping the electron in a circular orbit.
$$\frac{1}{4 \pi \epsilon_o}\frac{e^2}{r^2}=\frac{m_{electron}v^2}{r}$$
Solve for the speed of the electron.
$$v=\sqrt{\frac{1}{4 \pi \epsilon_o}\frac{e^2}{ m_{electron} r}}$$
$$v=\sqrt{\frac{(8.99\times10^9)(1.60\times10^{-19})^2}{(9.11\times10^{-31}kg)(0.53\times10^{-10}m)}}=2.17\times10^6 m/s$$
This is equal to 0.0072c, which is far less than 0.1c. The electron is not relativistic.