No, WebSocket API doesn't allow to set other header. Most headers other than Upgrade, Connection, Sec-WebSocket-* are not meaningful for the handshake, but I do agree that Set-Cookie would be a valid use case, as mentioned in the specification.
_The server can also set cookie-related option fields to set_cookies, as described in [RFC6265].
Thanks for the reply, jwaataws. Would love to see Set-Cookie allowlisted soon for Websocket APIs.
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