The paper develops the research area initiated in [\textit{R. Engelking}, \textit{R. W. Heath}, and \textit{E. Michael}, Invent. Math. 6, 150-158 (1968; Zbl 0167.20504)]. Let \(X\) be a non-Archimedean space in which every countable subset is closed. Then the following conditions are proved to be equivalent: (i) \(X\) is scattered; (ii) \(X\) is topologically well-orderable; (iii) there exists a continuous selector on the hyperspace of nonempty closed subsets of \(X\). The existence of continuous selectors for the hyperspaces of ultracompact scattered spaces and of spaces with a unique accumulation point is also investigated.