Drone Swarms are one of the enemy types in Duskers, and they are particularly nasty. To see what a Drone Swarm is, all you have to do is open the door to the unsafe room in Drone Operator Training.
You cannot outrun a drone swarm without upgraded speed or the speed boost module. Drone Swarms can sense you when stealthed if they pass over you. They will continue to attack you relentlessly even if you have a shield to protect you. There are, however, a few ways to deal with this situation in a way that will be beneficial to you.

If you have the teleport module, your best bet is to quickly teleport to a safe room as the drone swarm is about to attack or is attacking. Typing out "teleport r1" is a bit of a mouthful, though, so it's best to make a short alias like "qq" or "ee" that you can type quickly.

When you open an airlock, the room immediately becomes vacuum and all enemies and items are deleted on the spot. However, drones slide quickly toward the airlock and take damage as they scrape across the floor, and if the airlock is closed in time, the drone will survive. If the airlock is opened and closed on the same frame, e.g. by using "a2;a2" to instantly open and close the airlock, all drones in the room remain unharmed. This is useful to clear any kind of infestation from a room, but always has a chance of damaging doors in the room and any objects like generators or interfaces.

If you know you may have to do something risky in a room with an airlock, it's a good idea to make a temporary alias to instantly open and close the airlock in the event of an emergency. As with Option 1, give it a short name that you can type easily but not accidentally.

This also works when you have a shield upgrade - since you have more time, just drive to a room with an airlock and use this trick to get the swarm off you before your shield breaks. With a regenerating shield and a small swarm (e.g. with only a couple insects left after using turret) you have effectively infinite time to deal with the problem.

If none of the above options are not available to you, your drone will die. Period. But, before it dies, you can kill the drone swarm! In the Duskers help manual, you may have noticed the "destruct" command - this will disable your drone (as would the drone swarm) but it will also damage everything in the room, and it kills drone swarms every time.
However, it's a mouthful to type and requires being typed twice for confirmation. Best make an alias that you can type easily but not accidentally. The minimum time is 1 second, so use that.

Below is a demonstration: