Can the FMM be extended?
No — the FMM cannot be officially extended at INM. Once it expires, you must leave Mexico.
Visa runs
Many long-stayers do a visa run: fly out (Belize, Guatemala, the US) and re-enter, getting a new FMM. This works but is increasingly scrutinized — officers may grant short permits to visible repeat visitors.
Applying for residency
The legal long-term option. Apply for a Temporary Resident visa from a Mexican consulate ABROAD — you cannot start the process inside Mexico.
Risks
Overstaying your FMM means a fine of around $50 USD per month and possible difficulties on future entries. Consistent visa runs to live in Mexico can also start to flag the system.