So, maybe you were trying to mix mortar. Or there was a really stubborn bolt or screw that was stuck, and suddenly, your drill started smoking.
I’ve been there. Five years ago, I almost sacrificed my best cheapo Harbor Freight corded drill to mix this mortar:

If your drill is smoking, it’s likely the motor has overheated. Stop using the drill, and let the motor cool for at least 5 minutes. Then try again. If the motor continues to smoke, upgrade to a more powerful tool.
In my case, the motor started to smoke just as I finished mixing. I stopped, called it mixed, and five years later, the drill still runs fine!
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Why Your Drill Is Smoking
There are a couple other reasons your drill might be emitting smoking, which I’ll address below. However, by far the most likely reason is that the motor is overheating.
But why is your motor overheating?
If you’re asking your drill to do something difficult (like mixing mortar,) than that’s it. Basically, you’ve asked your drill to do something that’s beyond it’s capabilities, and you should probably upgrade tools if you want to keep doing that thing.
But if the drill is smoking during routine operation, then there’s something wrong. Here are a few possibilities:
Dull Drill Bit
If your drill bit is dull, the drill will need to work harder to cut through wood. This is especially true if you’re trying to drill through hardwood, which as implied by the name, is naturally harder.
The best way to check if you’re working with a dull drill bit is to replace the bit with a new bit, and see if the problem goes away.
If yes, then the problem was the drill bit.
Obviously, wait for your drill to cool before testing with the new drill bit. That way, you’re confident you’re testing the current bit (and not just seeing smoke from the already heated drill.)
Blocked Air Vents
If you look on the side of your drill, you’ll notice air vents (which is probably where the smoke is coming from!)
Like any vents, these can get clogged, especially since they tend to be around plenty of sawdust. This can cause hot air to be trapped by the motor, which will help the motor overheat.
If you have access to compressed air, your best bet is to blow air into those vents to clean them out.
Unfortunately, if you don’t an air compressor, your other options aren’t as great. I’ve tried using a blow drier before, which is a decent last ditch option, but the compressed air is much more effective.
Dying Drill
If your drill is reaching the end of its life, it will overheat faster and with less effort.
At this point, your best bet is to replace the drill, especially if this was a low quality drill to begin with.
If your drill was a nicer purchase and shouldn’t be dying so soon, it my be worth taking it to a repair shop. The brushes on brushed drills often wear out quickly, and can be replaced. A key sign that this is the case is if the drill is both smoking and sparking significantly.
(Note that a few sparks that stay inside the drill are normal. But if you see more sparks than usual AND the drill is smoking, your brushes are probably shot.)
However, high-quality modern drills are often brushless, so if your high-quality drill was a recent purchase, this shouldn’t be an issue.
When Your Tool Isn’t Powerful Enough
If you’re asking your drill to do something challenging, and that’s the primary reason it’s smoking, then chances are, you need a more powerful tool.
Lets consider what tool might be best:
Corded Drill
If you’re using a cordless drill to mix mortar, drill a large hole, or drill a hole though hardwood, a corded drill might be your answer.
All of these things require plenty of power, and sometimes a cordless drill simply isn’t enough.
I own a corded drill and two cordless drill, and I break out the corded drill anytime I need a lot of power to drill a hole.
My corded drill is a cheapo Harbor Freight one, costing $17. It’s a piece of crap, but I use it rarely enough that I don’t really care.
Impact Driver
If you were trying to drive or remove a screw and your drill started overheating, this is the tool you want.
(Note: If you driving the screw in, you drilled a pilot hole, right? Back the screw out, and drill a pilot hole before buying a new tool if you haven’t. You might also want to try a larger pilot hole as well before splurging on a new tool.)
Impact drivers exist solely to drive and remove screws, and they’re very good at what they do. They’re much more powerful than a typical drill driver, although they can’t be used to drill holes.
If you overheated your drill while working with screws, this is what you want.
Impact Wrench
If you were trying to tighten or loosen a bolt or nut, then the tool you’ll want to upgrade to is an impact wrench.
It’s like a impact driver, but intended for bolts and nuts instead of screws.
Other Reasons Your Drill Smells Smoky
So, if you can see smoke, then your drill has probably overheated.
But if something just smells funny… these other options are a possibility.
First off, if you’re working with a cordless drill, the battery can short. The easiest way to tell if the battery is the problem is to stick it on another tool. Still smells funny? It’s the battery.
Note that this is pretty rare, so I wouldn’t jump to this conclusion until you’ve ruled out everything else.
Alternatively, if something smells like it’s melting… it probably is. The drill casing is made of plastic, and as we all know, plastic melts. This is more likely with cheaper drill models.
In that case, your best bet is to let the drill cool off, then try again. If you still suspect some part of the drill is melting, replace (and maybe upgrade) your drill.