  We prove that every surjective isometry from the unit sphere of the space
$K(H),$ of all compact operators on an arbitrary complex Hilbert space $H$,
onto the unit sphere of an arbitrary real Banach space $Y$ can be extended to a
surjective real linear isometry from $K(H)$ onto $Y$. This is probably the
first example of an infinite dimensional non-commutative C$^*$-algebra
containing no unitaries and satisfying the Mazur--Ulam property. We also prove
that all compact C$^*$-algebras and all weakly compact JB$^*$-triples satisfy
the Mazur--Ulam property.
