No, it's not.

When lifting the other character you are either taking an action, or you interacting with an object -- depending on your DM's rulings.

If the DM rules that the pile driver is an attack (which I'd argue it is), that would cost you an action, and you can only repeat it if you have movement and an action to spend on it. This assumes you've grappled them, and have an attack left.

If the DM rules you are use an object. If it wasn't an enemy, you'd only be allowed to pick it up once. Once you drop it (which is free), it is no longer a free action to pick it up again. Use An Object action:

You normally interact with an object while doing something else, such as when you draw a sword as part of an attack. When an object requires your action for its use, you take the Use an Object action. This action is also useful when you want to interact with more than one object on your turn. (PHB 193)

Since you can't lift someone and do something else, it might take a full action to lift someone above your head. Remember that a round is 6 seconds, dropping someone more than once (or twice if you have the actions for it) would take about that long. Moreover, really can't lift someone and jump while doing something else, so it could be argued it can't be an interaction and must be a "Use Object" action.

To rule otherwise would allow a free 5d6 damage without using any action, and that, while awesome the first time (and a DM may allow it once, by Rule of Cool), is game breaking.

It's not in the rules, but DMs have the latitude to allow it.

As you point out, I couldn't find anywhere in the rules to support an action like that. However, the system allows for DMs to make judgments about whether an action is reasonable/possible, as well as how difficult it is (PHB 192):

When you describe an action not detailed elsewhere in the rules, the DM tells you whether that action is possible and what kind of roll you need to make, if any, to determine success or failure.

This specific instance is probably balanced.

Thus, it's up to you as the DM to determine whether this specific case should be allowed. More specifically, we need to ask, "If creatures are allowed to do this, would they always do it?"

From a theorycrafting perspective, it seems roughly balanced. The effects of pinning a caster's arms (or any character's arms, really), is roughly comparable to Hold Person, which is a second-level spell. We can take a cue from Hold Person, and say that if we want the effect to last multiple rounds, the grabbed character should get an opportunity to escape every round (as if from a grapple). The rest of the balance is probably a wash, because such a pin prevents the attacker from doing anything else, but they can do it for free.

We can also look to the grappler feat, which does something similar (PHB 167):

You can use your action to try to pin a creature grappled by you. To do so, make another grapple check. If you succeed, you and the creature are both restrained until the grapple ends.

Consider that you're allowing any character to take the benefit of this feat for free, and that the effect you're trying to create is actually more powerful than this feat. I'm personally OK with this, given that my players don't really use feats, but it's ultimately up to you and your players.

I have used a similar rule in actual play, and it seems to be a bit underpowered compared to other possible actions. I had a few of my NPC characters try to grab and pin the PCs, both to prevent them from escaping and to incapacitate them. I required my NPCs to initiate a grapple with one action, and then make a contested strength check to pin them with another action. Because the process costs two actions, allows for two "saving throws", does no damage, and still permits verbal-only spellcasting, its opportunity cost is very high--imagine the damage that a strong creature could do in two rounds! Additionally, once they are pinned, the grappler cannot do anything else, which effectively takes them out of the combat. While such an action was appropriate for the situation, I cannot imagine a smart combatant wanting to do this with any frequency.

However, if you do include this rule, you should remember that it will affect your future combats. It might not make much sense to try this move if there are lots of enemies, but it can completely shut down a combat with a solo spellcaster, if they ever move within melee range. It will also open up this option to enemies restraining the PCs, which they might not be super happy about.