It appears that zombies will not drop any items other than their basic drops when killed by the player.

They drop rotten flesh, carrot, potato, and iron ingot (and experience). They do not drop any armor or held items. I have tested this with a spawner on Hard difficulty in Survival mode (see attached screenshot).

I have tried a reset, and also a full uninstall/reinstall. I am not running any mods or packs or anything that isn't completely vanilla.