The authors establish a condition that guarantees isolation in the space of composition operators acting between \(H^p(B_N)\) and \(H^q(B_N)\), for \(0<p\leq\infty\), \(0<q<\infty\) and \(N>1\). The condition relates the extreme set of the self-map of \(B_N\). The isolation theorem is stated with the essential norm that is stronger than the general operator norm. This result in then used to characterize the component structure of the space of operators in certain cases, where \(0<q<p\leq\infty\).