an MP3 with the file's existing bitrate?
Yes, but you wouldn't want to. Every time Audacity makes a new MP3, it adds compression damage to the sound and you can't stop it. You can export as 320 quality MP3 and that helps a lot, but the file sizes go way up. Obviously, you can export as WAV and there is no further damage no matter how much editing you do.
So you were intended to create a master music file in WAV, AIFF or other uncompressed format, edit your brains out, export a new, different WAV and only then export your MP3. On the second pass, go back to either the master WAV or the edited WAV and make a new WAV -- a third one. Then export a new MP3. Never go back over old MP3s and never edit right over your old work -- or your master files. People who do that post here: "The lights went out while I was editing my two-hour podcast and it won't open any more. Please help, it's really important!"
We warn people never to do production in MP3 because MP3 causes sound damage and it's cumulative. The more you edit, the more damage.
You can set the MP3 export quality at the Export step -Options. You can make it anything you want, but you're not going to like staying at the same quality you started.
If you're doing very simple cuts and trims, you might want to look at a real MP3 editing program like MP3Split. The editing tools are limited, but they don't cause additional compression damage.
Koz