The answer is no, and the phrasing is key.
"assuming large quantities of reasonably priced materials" (are available)
Then yes, it would certainly be possible to manufacture a knife made from a single section of diamond. As @nick-t commented, Diamond knives exist
and are used in ultramicrotomy, the sectioning of a substance into extremely thin slices for use in transmission electron microscopy. The shaft is not diamond in this case, and the diamond blade is surrounded by metal as a mounting bracket, but these are solvable differences which mostly exist for cost reasons. Assuming large quantities of gem-quality diamond are reasonably priced and available, no problem.
This, however, isn't solvable:
...a functional knife...
Diamond doesn't create a functional knife. Ceramic knives made of aluminum oxide or other very hard materials exist, you have likely used them. They are not good (compared to steel) as they exhibit microchipping almost invariably. The same problem vexes the users of glass knives and obsidian ones; diamond knives are intended to delay that problem, and do, but this is in the hands of professionals doing surgery, or in a machine. In a normal, out-in-the-world usage scenario, a diamond knife would fail near-instantly. The crystal lattice of the diamond, which gives the diamond its hardness, also causes this fragility; so the problem is inherent to diamonds.
CATRA testers are widespread in the knife industry, at least among larger manufacturers. They are also widely known to be inaccurate in testing some steels, and especially ceramics; the problem arises because a human cannot cut as precisely as a machine can, and cannot entirely prevent lateral forces. A diamond knife would no doubt score a wonderful result in CATRA testing, as do ultrahard steels; but in reality, they would not be functional.