Sedimentary rocks in Akilia or at Hudson Bay have been proposed as the oldest in the world. My question is not related to rocks, but rather to sediment while still forming sedimentary deposits but before lithification (rock formation). While it might be difficult to put a clear limit between "sediments","lithified sediments" and "sedimentary rock", the question is what is the oldest sedimentary deposit on Earth still under sedimentary processes?
According to page 86 of Exploring the World Ocean:
In December 1989, ODP scientists drilled Hole 801C and recovered Jurassic-age rocks and sediments, about 170–165 million years old, from the Pigafetta Basin in the western Pacific, near the Mariana Islands. Sediments of nearly identical age had been found previously from the Deep Sea Drilling Project Hole 534A, located on the Blake-Bahama basin in the central Atlantic. These sediments and the rocks on which they rest are among the oldest oceanic crust found in the world ocean.
There is a more-detailed discussion at Site 801: Sedimentology and Biostratigraphy
The question is a bit confusing:
If (as suggested by the body of the question) you are interested in unlithified sediment only, then the transition from unlithified to lithified can be found at the bottom of most seas, lakes, and continental basins, but the factors determining the age or depth at which they became lithified rock are many, including the temperature/pressure/chemical conditions of the matrix. So far there is no way to determine the oldest unlithified sediment.
If (as suggested by the title of the question) you were not so interested in the lithified/unlithified bit, and assuming that by 'still under sedimentary processes' you refer to a place where sedimentation is still ongoing, then you are looking for the place on Earth where sedimentation has been active for the longest period down to the present. I believe we would call this 'the oldest active sedimentary basin' (with 'active' used in geology as 'presently accumulating sediment'). In such case, then the answer by DavePhD points to sediment deposited on the oldest subducting oceanic plate, I guess because oceanic environments should undergo nearly perennial sedimentation. But note that most of the sediment in that drilling is actually lithified, it became solid rock because it spent many million years at large pressures. Furthermore, the western Pacific is not the oldest oceanic plate on Earth. In the Eastern Mediterranean there are documented remains of the Tethys Ocean dating at 270 Myr. Sedimentation should have been uninterrupted ever since that plate formed. I have no idea if those sediments have ever been drilled.