This misconception about Welsh is slightly different, in that it tends to be propagated by well-wishers who actually defend the language (you can’t say that this website is entirely one-sided!).
Many supporters, with the best intentions, will often claim that “Welsh is the oldest language in Europe” (or one of). I wish they wouldn’t say this, because it’s untrue. It’s worse than that, in fact, because it’s not even wrong, in the sense that the claim itself doesn’t make sense.
It’s a bit like claiming that your family is older than someone else’s family. All families, and all languages, ultimately share a common ancestor, so really they’re all as old as each other.
Throughout history, parts of one family dynasty or clan have gone on to start another, yet at the time nobody would have noticed the difference. It’s the sort of thing that you can only observe with hindsight (rather like the emergence of new families of species in biological evolution). In the same way, the point at which a modern language is said to have sufficiently developed from its predecessor is rather arbitrary, and an academic convenience more than anything else.
It’s true that the language we know and love today as “Welsh” is said to have emerged from Brythonic at some point during the sixth century. But this hardly means that there was a point at which a parent was suddenly unable to understand her daughter (nor, for that matter, her granddaughter). Nobody going about their business in 550AD Wales (or the Old North up in modern-day Scotland for that matter) would have noticed anything unusual happening. Even though we call it Welsh from that point on, this certainly doesn’t mean that it’s not changed in the 1,450 or so years since.
Academics tend to subdivide its subsequent development into Primitive Welsh, Old Welsh, Middle Welsh, Early Modern and Late Modern Welsh. The gist of Middle Welsh (in which the tales of the Mabinogi were written) can sort of be understood by a modern reader, though not without some struggle. Anything older than that, however, is utterly incomprehensible, and might as well not be called Welsh at all. In practical terms, Old Welsh and Modern Welsh are not the same language.
Welsh is not even remotely a special case in this respect. Old English is said to be contemporaneous with Old Welsh. So is Classical Armenian. Old Swedish isn’t said to emerge until the 13th century, but that doesn’t mean that the Old Norse that was contemporaneous with Old Welsh is linguistically “further away” in any meaningful sense from modern Swedish than Old Welsh is from modern Welsh. These are just handy labels.
A wise man once said that a language is simply a “dialect with an army”. The distinction between two languages is political rather than scientific. Norwegian and Danish are classified as two separate languages even though they are much more similar than the versions of vernacular Arabic spoken in Mauritania and Oman, for example, which we merly call dialects. Again, my point is simply that these labels are mainly social constructs. I’m guessing that many of Europe’s languages claim to be among the continent’s “oldest”. Basque, famously, isn’t closely related to any other language in Europe and is said to be a survivor from the time before the Indo-Europeans arrived. Lithuanian, meanwhile, is sometimes called Europe’s most conservative language since it retains many ancient linguistic elements that others have long lost. Both, it may be conceded, would have a better claim to the “oldest language” title than Welsh, but it wouldn’t really be a linguistically meaningful thing to say in those cases either.
You may, very reasonably, ask whether any of this really matters. But even if it were true (which it isn’t), I would also argue that it isn’t actually a particularly helpful point for the language’s supporters to be making. Do we really want to imply that our language is some sort of curious fossil? Emphasising Welsh’s so-called antiquity plays into the hands of those who seek to dismiss the language as unfit for the modern age. Unfortunately, there are people who say that Welsh is a relic of the past that belongs in a museum. We should avoid making their case for them!