We introduce the notion of quasi-zero-dimensionality as a substitute for the notion of zero-dimensionality, motivated by the fact that the latter behaves badly in the realm of qcb-spaces. We prove that the category QZ of quasi-zero-dimensional qcblt;sub>0lt;/sub>-spaces is cartesian closed. Prominent examples of spaces in QZ are the spaces of the Kleene-Kreisel continuous functionals equipped with the respective sequential topology. Moreover, we characterise some types of closed subsets of QZ-spaces in terms of their ability to allow extendability of continuous functions. These results are related to a problem in Computable Analysis.