This is a great question with lots of interesting things to consider.   This appears to take us in a direction that considers the opposite of what R.R. Phelps found in 1957:

R.R.  Phelps, Convex sets and nearest points, Proc. Amer. Math. Soc. 8(4), 1957, 790-797.

I found a couple of early papers that might interest be of interest to followers of this thread:

B.G. Kelly, The convexity of Chebyshev sets in finite dimensional normed linear spaces, M.Sc. thesis, 1978:

https://www.math.auckland.ac.nz/~moors/Kelly2.pdf

For me, this is interesting it considers proximinal (semi-Chebyshev)  sets, p. 5.  A set is Chebyshev, provided it is both proximinal and semi-Chebyshev, i.e., if the proximinal set is a singleton for every x in the normed linear space X.   He proves Motkin theorem that every Chebyshev set is convex in $\mathbb{R}^2$, p. 22.    When Kelly wrote his thesis, it was known if Motzkin theorem can be extended from $\mathbb{R}^2$ to $\mathbb{R}^n$.    Perhaps one of the followers of this thread can pick up where Kelly (and Motzkin) left off.    Kelly himself does this on page 33, in R^n, every Chebyshev set is convex.

Another early paper of interest appeared in 1977 but not referenced by Kelly:

T.-C. Lim, Fixed point theorems for mappings of nonexpansive type, Proc. of the Amer. Math. Soc. 66(1), 1977:

http://www.ams.org/journals/proc/1977-066-01/S0002-9939-1977-0473935-9/S0002-9939-1977-0473935-9.pdf

A very recent paper speaks directly to the question for this thread:

Sangeeta, T.D. Narang, On the farthest points in convex metric spaces and linear metric spaces, Publications de l'Institut Mathematique 95(109), 2014, 229-238:

http://www.emis.de/journals/PIMB/109/n109p229.pdf

Additioanally a nice written Thesis of James Fletcher here:

https://www.math.auckland.ac.nz/~moors/James'thesis.pdf

Thanks are due to Prof. James F Peters (for his extremely insightful inputs) and Prof. Demetris Christopoulos for the useful thesis. Farthest point theory is significant in its own right as well as due to its connection with other branches and extremely important problems including the famous open problem on convexity of Chebyshev sets in Hilbert spaces (the thesis by James Fletcher, cited by Demetris, successfully solves the problem in inner product spaces). Farthest point theory also contains one of the longest standing open problems in Analysis, better known as the Farthest Point Problem(FPP), initiated by Prof. Victor Klee. The formal statement of the FPP may be stated as:

Is every closed uniquely remotal subset of a Banach space necessarily singleton?

The concept of Chebyshev center plays a very prominent role in the whole scheme of things and thus I believe this question is very significant. The geometric significance and intuitive feelings behind this problem is evident if we look at concrete examples of well-known geometric objects.

I would like to view a solution towards this problem as an important step towards understanding the norm geometry. Two points might be useful in the discussion:

1. The answer to the question is no (i.e., the center CAN NOT be the farthest point unless the set is singleton) if the space is an inner product space, in fact it is possible to characterize inner product spaces among normed linear spaces by using a norm inequality that proves my statement (Please see papers by Baronti, Casini and Narang on farthest points and uniquely remotal sets in Banach spaces).

2. If the space is not strictly convex then the center CAN be a farthest point. It is easy to see, just take a straight line segment which is part of the unit sphere (Since the space is not strictly convex, the existence of such of a non-trivial straight line segment on the unit sphere is always guaranteed. )

Thus it is even more tempting to speculate what happens if the space is strictly convex, which lies somewhere in between these two categories. It might also be possible that some other geometric properties of the space is the determining factor in whether or not the Chebyshev center can be a farthest point!

Dear Professor Debmalya Sain,

It seems to me that I can construct an example you are looking for. Enclosed please find the .pdf file with the construction.

Best regards, Valdimir Kadets

Respected Professor Vladimir Kadets,

Let me have the opportunity to thank you profusely for your contribution regarding this question. I would love to carefully follow your arguments at the first opportune moment. Yours is a well-known name and I am happy that you could find some time to think over our problem. I wish you a bright time ahead.

The elegant example furnished by Professor Vladimir Kadets successfully solves the problem. I thank him on behalf of every follower of this thread for coming up with such an original and non-trivial example!!!