  We define and study the metric, or extreme version of the notion of a
projective normed module. The relevant definition takes into account the exact
value of the norm of the module in question, in contrast with the standard
known definition that is formulated in terms of norm topology. After some
preliminary observations and discussion of the case where the base normed
algebra A is just the field of complex numbers, we concentrate on the case of
the next degree of complication, where A is a sequence algebra, satisfying some
natural conditions. The main results give a full characterization of extremely
projective objects within the subcategory of the category of non-degenerate
normed A-modules, consisting of the so-called homogeneous modules. We consider
two cases, `non-complete' and `complete', and the respective answers turn out
to be essentially different. In particular, all Banach non-degenerate
homogeneous modules, consisting of sequences, are extremely projective within
the category of Banach non-degenerate homogeneous modules. However, neither of
them is extremely projective within the category of all normed non-degenerate
homogeneous modules, when it is infinite-dimensional. On the other hand,
submodules of these modules, consisting of finite sequences, are extremely
projective within the latter category.
