Answer
$T$ is not independent of $v$
Work Step by Step
$r = \frac{\gamma~m~v}{\vert q \vert ~B}$
We can find an expression for the period:
$T = \frac{2\pi~r}{v}$
$T = \frac{(2\pi)~\frac{\gamma~m~v}{\vert q \vert ~B}}{v}$
$T = \frac{2\pi~\gamma~m}{\vert q \vert ~B}$
Note that $~~\gamma = \frac{1}{\sqrt{1-(v/c)^2}}$
Since $\gamma$ depends on $v$, then the period $T$ also depends on $v$
Therefore, $T$ is not independent of $v$
