The short answer is "no".

There have been claims of ancient tungsten artifacts. What appears to be, by far, the most prominent such claim is that of very small tungsten spirals (often called nanospirals) found in Russia. For example, these have featured in Hartwig Hausdorf's Wenn Götter Gott spielen (When gods play god), a von Danikenesque book claiming that humans evolved through extraterrestrial intervention, and also on the TV show Ancient Aliens, S11E12. However, these are variously claimed to be 15,000, 100,000, or 300,000 years old, and thus older than the period you ask about.

Oddly, those pushing these finds often cite a report by a Russian research institute, ZNIGRI, report No. 18/485 as evidence of possible extraterrestrial origin of these spirals. However, the report plainly states that

the age of the deposit (the riverbed) and the results of the tests give a very low probability to the assumption that the origin of these unusual, thread-shaped tungsten crystals is of a technogenic cosmic nature, due to the rocket take-off route from the Plesetsk space-station over the polar part of the Ural region.

That is, that they aren't extraterrestrial artifacts, but probably junk from terrestrial rockets. Hausdorf read this as meaning that they are of extraterrestrial origin, and claimed this "no" to mean "yes".

So, while these spirals have been claimed to be evidence of ancient tungsten artifacts, the only scientific investigation that appears to have been performed concluded that they are probably modern and terrestrial, and they can't seriously be considered as evidence of ancient tungsten.

In principle, tungsten could be smelted with pre-modern technology (but there is absolutely no evidence that this was done). The usual process is to heat tungsten oxide in a hydrogen atmosphere, at temperature of about 1000C, which is achievable with Medieval furnaces (I don't know the temperature required for smelting with carbon rather than hydrogen). However, this yields tungsten powder, and the melting point is prohibitively high for pre-modern melting of such powder to produce larger pieces (and even today, sintering with nickel is often the preferred approach, due to the high melting point).

References:

Supposedly a German translation of (parts of) the Russian ZNIGRI report: https://web.archive.org/web/20071117120919/http://www.freie-energie.net/index/raumschiffe/UFO-Funde%20im%20Ural/wolframspir_fund.htm