Asia
It's hard to think of the lapdog’s history in Asia without thinking of the Chinese Imperial House and their “lion dogs,” a dog so highly thought of it was illegal to own for anyone not part of the royal family.
Thanks to the exceptional records of the Chinese Imperial house, in this case the “Old Book of T'ang” that covers the history of the T’ang Dynasty (618-907), we are quite aware exactly when the lapdog arrived in the hands of Chinese royalty. The book details how two small pet dogs, “just six inches high”, were presented to the royal court of Tang-dynasty China in the year 624 by a merchant from the west. They were noted for their intelligence, impressing the ladies with their many tricks. This would be the start of what would become an almost thousand year-long history of lapdogs as an integral part of the imperial house of China.
It's stated that the merchant arrived from the eastern parts of the Roman Empire, making it not impossible that the dogs were melitan, or the descendants of such. However it's also possible he bought them from someone on the many many miles long trip, and they hadn't been in his possession for that long.
The next mention of a lapdog in Chinese history is found in the retelling of how the emperor Tang Xuanzong (ruling between years 712-756) is playing a board game with a prince while one of his concubines watches with a dog on her knee. Realising the emperor is about to lose, the concubine releases the dog who runs over and flips the game, to the emperor's delight. By then, the lapdog had already found a spot in the heart of the Chinese royalty, a spot it wouldn’t release until the fall of the imperial house in 1911. It's around this time we have the first chinese art-pieces portraying small pet dogs.
T’ang-Dynasty (Year 608-907)

“Court Ladies wearing flowers in their hair”, T'ang Dynasty, 8th Century, Zhou Fang.
Liaoning Provincial Museum.

Details from “Court Ladies wearing flowers in their hair”

Detail from "Seated Court Lady", T'ang Dynasty China, 8th Century.
Metropolitan Museum of Art.
11th Century

"Palace ladies bathing children", T'ang Dynasty China, 11th Century, Unidentified Artist.
Metropolitan Museum of Art.

Detail from "Palace ladies bathing children"

Detauk from “Tea Ceremony” mural, Liao-Dynasty China, 11-12th Century. Xiabali Liao Dynasty tombs.
Xiabali Village, Xuanhua, Hebei.
1100s

“Hibiscus and Rocks” part of a collection of paintings, Song Dynasty China, 12th Century. Li Ti.
The National Palace Museum, Taipei City

“Daylilies and Dogs at Play”, Nara period Japan, 12th century, Mao Yi (毛益).
Private collection, Japan.
1400s

明 佚名 犬戲圖 冊頁 (“Puppies Playing beside a Palm Tree and Garden Rock”), Ming Dynasty China, 15th Century. Unknown Painter.
The Metropolitan Museum of Art, NY

“Dog and Bamboo”, Ming Dynasty China, 1427, Zhu Zhanji (1399 - 1435).
Nelson Atkins Museum of Art

“Garden Scene”, Ming Dynasty China, mid 15th Century. Shang Xi (商喜).
The National Palace Museum, Taipei City
1600s
1700s

Detail from “Courtesan and Attendant Playing with a Dog”, ca 1716-1736, Nishikawa Sukenobu (Japan).
Met Art Museum, NY, USA.

Detail from “Painting of Lady and Children”, Quing Dynasty China, first half of 18th century.
Private Collection.

Detail from "An Oiran Holding a Black Dog", 1769, Suzuki Harunobu (Japan).
Met Museum of Art, NY, USA.

Detail from “A Young Woman with a Dog”, ca 1790-91, Torii Kiyonaga (Japan).
Met Art Museum, NY, USA..

“Dog and Peonies”, Edo period Japan, late 18th to early 19th century, Kaneko Kinryō.
The Seikado Bunko Art Museum, Japan.
1800s

"Dutch Woman Holding a Dog", early 1800s, unknown japanese artist.
Met Museum of Art, NY, USA.

"Surimono Calendar for the Dog Year, 1814", 1814, Totoya Hokkei (Japan).
Met Museum of Art, NY, USA.

“畫犬插屏”, Dog Painted on a Table Screen, Quing Dynasty China, mid 19th century, unknown painter.
The National Palace Museum, Taipei City.

Painting of a Dog on a scroll, Quing Dynasty China, late 19th century, Shen Zhenlin (沈振麟).
National Palace Museum, Taipei City.

Painting of a Dog on a fan, Quing Dynasty China, Li Tingliang (李廷樑).
National Palace Museum, Taipei City

Details of scene of the Quing Dynasty Court, late 19th century, unknown court painter.
National Palace Museum, Beijing.

Details of scene of the Quing Dynasty Court, late 19th century, unknown court painter.
National Palace Museum, Beijing.

“粉彩花卉名犬瓷板”, Painted Enamel Plaque, Quing Dynasty, unknown painter.
The National Palace Museum, Taipei City.

“粉彩花卉名犬瓷板”, Painted Enamel Plaque, Quing Dynasty, unknown painter.
The National Palace Museum, Taipei City.