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Metals form metallic bonds which explain many of their chemical and physical properties.

This is most familiar in the solid state but metallic properties are still quite recognizable in the liquid state e.g. mercury and molten iron.

However, gaseous metals are much less familiar and it is not obvious whether they would still have a distinct metallic behaviour. For example, is there a distinction between gas and plasma for a metal?

This article mentions that the metallic bond persists in the liquid state but says nothing about the gas state: Metallic bonding at Chemguide.

So, do metals retain any distinct metallic behaviour in the gas state?

Note that I am not asking only about the electrical properties. My main question is whether the gas consists of neutral atoms, small molecules, or positive ions in a sea of electrons?

No, gaseous metals do not retain metallic bonds, nor metallic conductivity, nor luster, nor any other metallic properties. They are no different from other gases. True, they typically require pretty high temperatures to form, but then again, they are hardly special in this regard, as many non-metallic substances require the same.

See, all metallic properties are in fact collective effects. They are caused by metallic bonding, and not just of two atoms, but of an entire piece of metal (like a thousand, only much more). You don't have bonding in gases. The particles (lone atoms in this case) are basically free to go. Maybe they bump into each other, two or three at a time, but definitely not a thousand.

Plasma is an entirely different thing, unrelated to your initial question. Yes, plasma is pretty different from a gas; to obtain it, you'll have to heat the gas until it ionizes.

So it goes.