In 1892, the Hongkong Rope Manufacturing Company set an application "for the construction of a single line of Tramway in Davis Street from the Rope Manufactory to the Praya." It looks that nothing happened for almost 10 years.
Next is the Ordinance 36 of 1901
An Ordinance to authorise The Hongkong Rope Manufacturing Company, Limited, to construct a Tramway within the Colony of Hongkong
It is referred there:
A partly single and partly double line commencing at a point (shown on the said plan [not available]) near to and to the South of the North-east corner of Inland Lot No. 906, thence passing (by a curve) in a Northerly direction along Smithfield and across the Praya in the position shown on the said plan and terminating upon the wharf intended to be erected in the position shown on the said plan.
Probably the tramway started its operation in 1902 or 1903. It can be seen on the 1922.1 map on HKMAPS.HK
The book "Twentieth Century Impressions of Hongkong, Shanghai, and other Treaty Ports of China" from 1908 refers:
The raw material is conveyed on a private trolley line from the wharf to the storage godown…
The ordinance was renewed in 1913 for another 10 years.
From 1924 onwards, the duration of rights was extended for one year, last in 1941.
During WW2, the production likely stopped and was not resumed after the war. So the end of operation was guessed to be end of 1941.