My daughter who is 4 was born in the UK and holds a British Passport. Her mother who is Bulgarian wants her to have a passport from Bulgaria. Question is will she be able to retain the British passport if she has a second passport.
Yes, your daughter will be able to retain both her passport and her British citizenship.
Dual citizenship (also known as dual nationality) is allowed in the UK. This means you can be a British citizen and also a citizen of other countries.
Source: Dual Citizenship
Bulgaria allows its citizens to hold foreign citizenship in addition to their Bulgarian citizenship. Some countries, however, do not permit multiple citizenship e.g. adults who acquired Bulgarian and Japanese citizenship by birth must declare, to the latter's Ministry of Justice, before turning 22, which citizenship they want to keep.
Unlike some other countries, the UK places no restrictions on the number of nationalities a person can claim. There are an estimated 500,000 dual or treble nationals inside the London Orbital alone (my own family, for example, is treble).
She can also use her Bulgarian passport to come and go from the UK, but the rules on that are more complex.
The controlling legal reference is the British Nationality Act 1981