Yes you could generate a rsa key pair in the browser and send a tuple of user id (or just a nonce) and public key to the server as a form of automatic registration. The client could prefix each chat with the user if/nonce and sign it with the private key before sending it to the server. From then on the server could simply retrieve the public key it has associated with the user id prefix and reject any messages that fail signature validation.

This could also work in a peer-to-peer context by only using the server for public key registration (i.e. by chat room). All messages would go directly between clients and the server would never receive chat messages.