[pjc50]

I'm not aware of any open basebands existing, and I'm not sure how they'd pass regulatory requirements.

There is an open GSM base station system though, OpenBTS. But you definitely can only use that for testing.

[Gregordinary]

The FreeCalypso Project is working on a libre phone. Looks like they're using a baseband processor from a TI Chipset.

https://www.freecalypso.org/

[Sir_Cmpwn]

http://osmocom.org/projects/baseband