I very much doubt an extension of
an RP can be done in China, or even Hong Kong.

The system of extending RPs/tourist visas/business visas was discontinued around a decade
ago.

Residential permits, like all visas, can be extended at the discretion of the local Immigration police. It must be done within the province that issued the RP. Thirty days is the customary extension. The 'extension' requires the cancellation of the existing RP and the process must begin at least 7 days prior to the RP's date of expiration. The '30 day extension' begins the day that the passport is handed in, so that the actual extension is no more than 23 days. A new 'visa' is added to the passport which merely states the date that the holder must exit China.