  Suppose $L(H)$ is the space of all bounded linear operators on a complex
Hilbert space $H.$ This article deals with the problem of characterizing the
extreme contractions of $L(H)$ with respect to the numerical radius norm on
$L(H).$ In contrast to the usual operator norm, it is proved that there exists
a class of unitary operators on $H$ which are not extreme contractions when the
numerical radius norm is considered on $L(H).$ Moreover, there are non-unitary
operators on $H$ which are extreme contractions as far as the numerical radius
norm is concerned.
