Tsim Sha Tsui Market [c.1911-c.1999]
Appears first as a red square on the 1904 map (on hkmaps.hk), is labelled "MARKET" on the 1910 map. Location is on Peking Road (former Chater Road) near the junction with Canton Road.
PUBLIC WORKS DEPARTMENT. No. S. 144.—It is hereby notified that tenders be received at the Colonial Secretary's Office until Noon of Monday, the 15th August, 1910, for the erection of Shops, Stalls, &c., at the New Market, Tsim Sha Tsui, Kowloon. .....
So the market probably opened end of 1911.
On the 1965 map it's still there and labelled "M".
Update 2020-01-08: The building stood there till the end of the 1990's (see here). Construction of "One Peking Road" started in 2000, so demolition date was assumed to be 1999. From the information found in the old Gevernment reports, the correct name of the market was not "Peking Road Market"(as used for this place so far) but "Tsim Sha Tsui Market".
Update 2020-04-01 (from the comment below)
1978: The stalls inside and outside the market moved to the temporary market of Haiphong Road.
March 5, 1979: The market building was remodeled and used as the Tsim Sha Tsui Post Office.
1981: The market building was converted into a police recruiting center and later a shopping mall.
1999: Demolition of the market.