Yes a black mamba can kill an elephant with it's venom.
A black mamba snake can deliver roughly 300–400 milligrams of venom (in several bites) so if the black mamba snake bites the elephant more than once then it could introduce enough of the venom into the elephants blood that it kills it.
The black mamba snake would not be able to kill the elephant in a single bite though as it would take at least 3 to 4 bites from the black mamba for the elephant to die of the snake bite.
It takes around 500 milligrams of snake venom from the black mamba snake for the elephant to be affected and killed by it.
So if the black mamba does not introduce enough venom into the adult elephant or any elephant then the elephant will likely not die.
The black mamba snake also known as Dendroaspis polylepis snake, is a species of large, extremely venomous snake belonging to the family Elapidae.
The black mamba snake is native to parts of sub-Saharan Africa.