First, you need to differentiate between the WebSocket protocol and the WebSocket API within browsers.
The WebSocket protocol has a frame-size limit of 2^63 octets, but a WebSocket message can be composed of an unlimited number of frames.
The WebSocket API within browsers does not expose a frame-based or streaming API, but only a message-based API. The payload of an incoming message is always completely buffered up (within the browser’s WebSocket implementation) before providing it to JavaScript.
APIs of other WebSocket implementations may provide frame- or streaming-based access to payload transferred via the WebSocket protocol. For example, AutobahnPython does. You can read more in the examples here https://github.com/tavendo/AutobahnPython/tree/master/examples/twisted/websocket/streaming.
Disclosure: I am original author of Autobahn and work for Tavendo.
More considerations:
As long as there is no frame/streaming API in browser JS WebSocket API, you can only receive/send complete WS messages.
A single (plain) WebSocket connection cannot interleave the payload of multiple messages. So i.e. if you use large messages, those are delivered in order, and you won’t be able to send small messages in between while a big message is still on the fly.
There is an upcoming WebSocket extension (extensions are a builtin mechanism to extend the protocol): WebSocket multiplexing. This allows to have multiple (logical) WebSocket connections over a single underlying TCP connection, which has multiple advantages.
Note also: you can open multiple WS connections (over different underlying TCPs) to a single target server from a single JS / HTML page today.
Note also: you can do “chunking” yourself in application layer: send your stuff in smaller WS messages a reassemble yourself.
I agree, in an ideal world, you’d have message/frame/streaming API in browser plus WebSocket multiplexing. That would give all the power and convenience.