Do we have a peak in an overdamped RLC circuit?
An overdamped RLC when "hit" with a step change input does not produce any overshoot: -




  I got the equation for the natural response and differentiated it to
  get a maxima, and the time corresponding to it is the time for the
  first peak.


You used the wrong equation then. There are three typical equation examples from this site: -



You need to use the over-damped equation. And note that only the underdamped equation produces ringing.


  But my instructors said that there's no peak in case of an Overdamped
  RLC circuit? Is he right?


Yes, he is correct.