To the best of my knoeledge, the problem remains open in the more concrete setting of linear operators between Banach spaces, endowed with the usual operator norm. It may sound strange, but even the finite-dimensional case remains open! I have personally studied the problem, along with some co-authors, exclusively in the Banach space context, and mostly in the finite-dimensional case. I think that it is a particularly hard problem, even in this far more restricted scenario.

Without a shadow of doubt, the problem deserves far more attention from the functional analysis community, and therefore, I must thank you for focussing on such a simple yet deep question.

Having said all these, I do know of some interesting results by Lindenstrauss and Perles, Lima, Sharir, T. S. S. R. K. Rao among others on this topic. It might be fruitful to look into their ideas and results (again, mostly in the Banach space context). We have also tried to contribute something to this intriguing topic. We (Paul, Mal, Sain) we have obtained a complete geometric characterization of extreme contractions between two-dimensional strictly convex and smooth Banach spaces. We (Paul, Ray, Roy, Bagchi, Sain) have also been able to obtain some interesting conclusions on extreme contractions between polyhedral Banach spaces. Recently, with Paul and Sohel, we have also obtained a complete characterization of extreme contractions between finite-dimensional polyhedral Banach spaces. Most of these papers are available on RG and please feel free to look at them, should you choose to!

Of course, much remains to be done on this fascinating question, and from multiple points of view, including analytic, grometric and combinatorial. In short, a potential exciting journey into the unknown!