There are apparent examples of languages being invented, with no clear parent language. [1] When education for the deaf is first introduced, children from many different families are brought together. Most do not have full language - they cannot communicate with language with anyone. Some are part of small clusters, families or small villages where many are deaf. [2] They often do have full languages. These languages are usually untraceable in origin. They're not related to each other, usually.
Pool these kids these together in a school and they will create a full language, a creole [3], in just a few years.
It's unclear, IMO, exactly how to interpret that effect. The children are not completely without language. They're obviously aware of the concept of language, a few of them may even have full languages (even if they can't use it with the other kids) and the teachers are also trying to teach them language, usually American sign if there's nothing local. So it can't be said to be language popping up in a true vacuum. But it is certainly curious.
[1] https://en.wikipedia.org/wiki/Nicaraguan_Sign_Language
[2] https://en.wikipedia.org/wiki/Martha%27s_Vineyard_Sign_Langu...
There are a few more concrete reasons to specifically believe against independent development of language. If you believe that language faculties arise very quickly after evolution of a biological organ to facilitate language, that suggests that the ability to have language is a key part of Homo sapiens and therefore language would have originated only once in the development in Homo sapiens as a species.
If this is the case, then the first language would have probably rapidly spread between the early populations in africa – if, indeed, there were multiple distinct populations at the time – and new (homo sapiens–originated) languages would have only arisen if the chain of speakers were somehow broken and children raised without access to an existing language.
With that being said, there are some cases where something similar has happened. Most obviously are sign languages, which are believed to have had multiple origins, since it is possible for deaf people to be born among non-signing populations. Something similar in the case of spoken languages is the phenomenon of pidgins evolving into creoles mentions elsewhere in the comments, although they still retain at least some aspect of one of their parent languages. The extent to which the latter count as proto-languages is questionable.