Like other world governments, Taiwan has publicly congratulated Trump on his victory in November's presidential election as it seeks to get onside with the next US leader.
The United States does not have official diplomatic relations with Taiwan, but Washington has long been Taipei's biggest backer and provider of arms.
Trump caused jitters during his campaign by suggesting Taiwan should pay the United States for its defence and accusing the island of stealing the US semiconductor industry.
"Taiwan is confident that it will continue to deepen cooperation with the new government to resist authoritarian expansion, and create prosperity and development for both countries while making more contributions to regional stability and peace," Lai told reporters in Palau.
Lai arrived in the tiny Pacific island nation on Thursday after visiting the American territory of Guam where he spoke with Johnson -- the highest-level US contact the Taiwanese leader has had during his week-long trip.
China claims Taiwan as part of its territory and opposes any international recognition of the island. Beijing especially bristles at high-level official contact between Taipei and Washington.
A 2022 visit to Taiwan by then US House speaker Nancy Pelosi prompted China to launch military drills around the self-ruled island.
China's foreign ministry on Friday warned Taiwan that "seeking independence with the help of the United States will inevitably hit a wall", and called on Washington to "cease meddling in Taiwan-related affairs".
In response to a question about possible Chinese military drills around Taiwan this weekend, Lai said "raising your fists is not as good as opening your hands".
Lai also insisted that Taiwan and China were "not subordinate to each other".
"No matter how many military exercises, warships and aircraft China sends to coerce neighbouring countries, it cannot win the respect of any country," Lai said.
- 'Rock solid' relations -
Lai's Pacific tour -- his first overseas trip since taking office in May -- is aimed at fortifying ties in the Pacific where China has been poaching its allies.
Palau is among 12 nations that still recognise Taiwan's claim to statehood, after China convinced others to sever diplomatic relations with Taipei in favour of Beijing.
Earlier, Lai and his Palau counterpart Surangel Whipps Jr watched a joint rescue exercise involving the Taiwan's largest coast guard patrol ship and two vessels donated by Taiwan to Palau.
Before that, Lai attended a ribbon-cutting ceremony for a new "one-stop" government services building in Palau that Taiwan helped fund.
Lai hailed the building project "a model of successful bilateral cooperation" and said the Taiwan-Palau alliance was "rock solid".
- 'No winners from conflict' -
The dispute between Taiwan and China goes back to 1949 when Chiang Kai-shek's nationalist forces were defeated by Mao Zedong's communist fighters and fled to the island.
While Taiwan calls itself a sovereign nation, with its own government, military and currency, Beijing insists the island belongs to China and has not ruled out the use of force to bring it under its control.
Taiwan faces the constant threat of a military attack by China, which regularly deploys fighter jets and warships around the island to press its claims, and it relies heavily on US arms sales to boost its defences.
On the eve of Lai's Pacific tour, the United States approved a proposed sale to Taiwan of spare parts for F-16s and radar systems, as well as communications equipment, in deals valued at $385 million in total.
Speaking during a two-day visit to the US state of Hawaii on Saturday, Lai said there was a need to "fight together to prevent war", warning there were "no winners" from conflict.
Lai returns to Taipei on Friday, wrapping up a trip that also included visits to Taiwan's other Pacific island allies the Marshall Islands and Tuvalu.
bur-amj/tym
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