If you come to Taiwan via "visa-free" or "visa-on-arrival,” in general, you cannot extend the period of stay. However, you can apply for a visitor visa at the Bureau of Consular Affairs of the Ministry of Foreign Affairs within 5 working days before the expiration of your visa if the following conditions make it impossible to leave the country as scheduled:

1. Serious illness

2. Natural disaster or other force majeure accidents.

(Article 4 of the Enforcement Rules for the Issuance of ROC Visas to Foreign-Passport Holders)