This is a question I wanted to ask for some time now. You learn in solid state theory that the free electron model is the reason for metals conducting electric current. The electron orbitals delocalize and electrons can move freely in the metal ion lattice. This model seem to work well for solid metals, but there are liquid metals and they seem to me to conduct electric current in the same way as their solid cousins. But in a liquid the atoms are moving around similar to a gas, but with some more scattering between atoms. It this incorrect for a liquid metal? I guess it is; there has to be free electrons there as well. Thus is a liquid metal really an ion liquid?
Even in liquid metals, the current is mostly carried by electrons, ions are just too heavy in comparison to play a major role in conduction (though the situation is different in polar liquids).
The difference between electrons in a solid (e.g. copper wire in your phone) and in a liquid metal is that the scattering rate in a liquid is very high, so usually the resistance goes up, though for some poor metals such as Bismuth, this isn't true.
Usually metals have large bandwidths (on the scale of electron volts), and so even for temperatures above 1000K you can expect the bonding of electrons to be the delocalized and mostly in tact. In this sense, the conduction is not too different from an amorphous metal, where electrons are delocalized, but there is strong disorder scattering. The timescales for liquids and amorphous metals is quite different with regards to ionic motion, but as far as being delocalized electronically, they are rather similar.
Long story short, the electrons in liquid metals are generally delocalized, just like when they are in solids. The difference is in the electron-ion scattering interactions, which are typically stronger for liquid metals to their high temperature and lack of ordering. Experimental proof for this claim is that even liquid metals obey the Drude form for the optical response (although you need to adjust electron-ion interactions appropriately), which means the electrons are nearly free.
I don't think that your analogy to compare the liquid metals with a gas is correct. In a gas, the individual atoms are far away from each other for them to interact in any way(except when they are in near vicinity). In a liquid, there are always electron cloud interactions with the neighbouring atoms which are responsible for it to stay in the liquid phase.
Considering the case of liquid metals, the metal atoms have enough energy to free some electrons which can participate in conduction. Moreover, the mechanism of conduction, in this case, is totally different as the ions are no longer stationary and the electrons will more often bump into the atoms. Thus, the liquid metals cannot conduct current in the same way as their solid counterparts.