With the release of the new Mordenkainen Presents: Monsters of the Multiverse, the revamped Fire Genasi trait "Reach to the Blaze" now allows the race access to the flame blade spell:
[...] Starting at 5th level, you can also cast the flame blade spell with this trait, without a material component. [...]
This made me think, if flame blade can be used with the paladin's Divine Smite feature, casting flame blade at 5th level with Divine Smite using a 5th-level spell slot will result in 4d6 fire damage and at least 5d8 radiant damage.
Does flame blade count as a melee weapon that can be used with Divine Smite?