Is metallic sodium a strong base?
No, stricly speaking metallic sodium is not a strong base. Strictly speaking, metallic sodium is not a base at all. (Sodium amide clearly is one, though: due to the amide part.)

Not everything that reacts with an acid is a base. If an acid reacts to form hydrogen and there are no hydrides present, that is a very strong indicator that the reaction was not an acid-base reaction but a redox reaction. And that is what metallic sodium is: a strong reducing agent.

Just because reducing agents can react with acids (proton donors) to generate hydrogen radicals and thus nascent hydrogen gas does not make them bases. For the reaction partner of an acid to be a base, it must accept (and keep!, at least to a certain extent) the proton donated by the acid — or, in the Lewis theory, it must donate an electron pair.