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We define and study the metric, or extreme version of the notion of aprojective normed module. The relevant definition takes into account the exactvalue of the norm of the module in question, in contrast with the standardknown definition that is formulated in terms of norm topology. After somepreliminary observations and discussion of the case where the base normedalgebra A is just the field of complex numbers, we concentrate on the case ofthe next degree of complication, where A is a sequence algebra, satisfying somenatural conditions. The main results give a full characterization of extremelyprojective objects within the subcategory of the category of non-degeneratenormed A-modules, consisting of the so-called homogeneous modules. We considertwo cases, `non-complete' and `complete', and the respective answers turn outto be essentially different. In particular, all Banach non-degeneratehomogeneous modules, consisting of sequences, are extremely projective withinthe category of Banach non-degenerate homogeneous modules. However, neither ofthem is extremely projective within the category of all normed non-degeneratehomogeneous modules, when it is infinite-dimensional. On the other hand,submodules of these modules, consisting of finite sequences, are extremelyprojective within the latter category.

Metric version of projectivity for normed modules over sequence algebras