TAIPEI, May 13 (Reuters) - Taiwanese cities are tightening restrictions on access to public venues like gyms and libraries after a rare rise in domestic COVID-19 infections that has spooked the stock market and unnerved the island.

Early and effective prevention steps, including largely closing its borders, succeeded in shielding Taiwan from the worst of the pandemic. It has reported just 1,231 infections so far.

But markets and the government have been on edge since renewed domestic outbreaks began late last month, with 16 new domestic cases announced on Wednesday setting a record daily high. read more

The benchmark index (.TWII) fell around 3% on Thursday morning, though also tracking losses in the United States overnight.

New Taipei City, which surrounds the capital Taipei and where some of the new cases have been reported, has ordered libraries and internet cafes closed, while Taipei has ordered a limit of 100 people at a time in gyms.

Taipei Mayor Ko Wen-je, writing on his Facebook page late on Wednesday, said he knew people were starting to worry, but that there was no need.

"Please don't panic, don't get into an upheaval, Taiwan has abundant medical capabilities, and the number of severely ill patients has not increased rapidly," wrote Ko, who is a medical doctor himself.

The government, which has a robust contact tracing and quarantine system, is racing to locate the source of some of the local infections, concentrated in densely populated northern Taiwan.

Chen Chi-mai, the mayor of the southern metropolis of Kaohsiung, home to a major port, said authorities were disinfecting locations where one of the cases had visited to make offerings at a temple and were quarantining and testing 56 contacts.

The central government has already suspended mass gatherings of more than 100 people inside and 500 people outside until next month, and warned it may take further steps that could close non-essential businesses.

Taiwan has never gone into a complete lockdown.