Replacing the broken part
You certainly don't need to buy a whole new extruder assembly, including a stepper motor.
You could just buy a new extruder mechanism, and upgrade from the plastic1 extruder, to a metal (aluminium(?)) extruder, such as the one (apparently) sold by Creality themselves, such as this2, Metal Extruder Kit (Red):

However, if you just need a certain piece of the extruder (from your photo, I can't tell which part is broken), you could search for that, although, it would probably be cheaper just to buy the whole thing.
As to the name of the particular part that is broken, again, it is difficult to tell from the photo which part is actually broken. If you could add an annotated photo or more explicitly show the broken part, then it would be easier to provide a name of the actual part.
That said, after re-reading your question, I think that you may be referring to the whole black assembly - along with the spring, brass knurled nut, pulley/bearing, and associated bolts - that sits on the stepper motor... if so, then that is the extruder.
Identifying the parts of an extruder
It can be confusing knowing exactly what the names of the various parts of the extruder are - seeing as the entire extruder assembly (including the hotend and stepper motor) is often referred to as the extruder, either individually or as a whole.
Here are a couple of images that map out the various parts of the extruder assembly:
From the excellent RepRapWiki - Extruders

From Extruders 101: A crash course on an essential component of your 3D printer

Note that:
- The hobbed bolt, is sometimes referred to as the knurled bolt or knurled nut
- The idler wheel is (generally) just a bearing
Footnotes
1 As confirmed in the comments, the extruder that comes with the non-pro version of the Ender 3 is plastic.
2 I'm not recommending this seller, nor am I sure that they are genuine Creality parts... this is just an example for what you need to be looking for.