The Holy See is the only European state that still has diplomatic relations with Taiwan. "Although relief efforts in Hualien are more than sufficient, your warmth & kindness have been felt by the people of Taiwan", Tsai tweeted.
Tsai said she "ordered search and rescue workers not to give up on any opportunity to save people, while keeping their own safety in mind".
Council officials said the foundation had passed on the information about the people killed to its mainland counterpart, the Association of Relations Across the Taiwan Strait (Arats). It was said to have caused four buildings in Hualien city to tilt or even cave in, reported ABC News.
Rescue workers have saved one person from a hotel that was severely damaged after a strong quake hit near Taiwan's east coast. Rescuers said five other mainland tourists are still trapped in the damaged Yun Men Tusi Ti building, including four adults and one child.
Japanese Chief Cabinet Secretary Yoshihide Suga said his country was dispatching a rescue team to help in the search effort.
In September 1999, a 7.6-magnitude quake in Taiwan killed more than 2,400 people and left 100,000 homeless.
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With aftershocks continuing to hit after the quake, residents were directed to shelters, including a newly built baseball stadium, where beds and hot food were provided.
Progress in the search and rescue operations has been slow at the worst-hit building in Taiwan's Hualien County, where a 6.5-magnitude natural disaster has killed 10 and injured over 260 others.
Yu's husband arrived in Taipei on Wednesday, the foundation said, adding that it was helping Yu's mother and brother, as well as the families of the three other women killed to travel to Taiwan to retrieve the bodies. Seven of the dead perished in that building and the remaining missing are believed to be hotel guests there.
Taiwanese broadcaster TVBS said Wednesday that a rescue operation was underway at the Marshal Hotel.
China Central Television reported that more than 40 of the missing people were trapped in the Yunmen Cuiti building, a 12-story apartment building. While most of the unaccounted for were likely to be in one residential building, the agency did not have an estimate of how many were in the building at the time. "So we can't use large machinery for fear that the people there might be harmed if the machinery happens to damage or collapse the structure", said rescuer Wang Ming-chung. The force of the tremor late Tuesday buckled roads and disrupted electricity and water supplies to thousands of households.