  Let $f: S(E) \to S(B)$ be a surjective isometry between the unit spheres of
two weakly compact JB$^*$-triples not containing direct summands of rank
smaller than or equal to 3. Suppose $E$ has rank greater than or equal to 5.
Applying techniques developed in JB$^*$-triple theory, we prove that $f$ admits
an extension to a surjective real linear isometry $T: E\to B$. Among the
consequences, we show that every surjective isometry between the unit spheres
of two compact C$^*$-algebras $A$ and $B$ (and in particular when $A=K(H)$ and
$B=K(H')$) extends to a surjective real linear isometry from $A$ into $B$.
These results provide new examples of infinite dimensional Banach spaces where
Tingley's problem admits a positive answer.
