consider me sitting on the top of a train which is travelling close to the speed of light, will I be able to see my image on a mirror which I'm holding in my hand??
Yes, as per the principle of relativity.
This is precisely the sort of thought experiment Albert Einstein started out with. It turns out that yes, you will be able to see your image in the mirror when you move close to the speed of light. You will also not notice anything strange about that image, or anything strange about things that are moving with you in your local reference frame.
This might seem strange in the sense that the rays of light will appear to take a much longer time to reach the mirror, and a much shorter time to be reflected back to the moving observer, when looking at that observer and his mirror from an inertial reference frame "at rest".
This "strangeness" is easily resolved though, if you give up the idea that time is some sort of omnipresent thing which both observers always agree on. This, as special relativity has shown (and general relativity elaborated on), is simply not true for the universe we live in; if you start moving, we will start disagreeing on how time works (but still both be correct).
The main point is that you cannot travel at exactly the speed of light. You can reach 99%, or even 99.999999% (theoretically, of course; you'd be torn apart in reality). So you will still see your reflection, and it will look the same since the mirror is in your own inertial frame.