The language of all borderless words Could anyone help me please to find who was the first person who has proved that the language of all borderless words is not regular and when was that? Could you mention the reference, please?

A word $w$ has a border $u$ if $u$ is both prefix and suffix of $w$ (and $u$ does not equal $\lambda$ or $w$).
The borderless words are also known as unbordered or bifix-free ("A note on bifix-free sequences", PT Nielsen, in a paper as early as 1973).

The language of unbordered words is not even context-free, see the paper "Inverse star, borders, and palstars" by Ramparsad etal, in TCS with a preprint on ArXiv. The technique to obtain this result is Ogden's Lemma, an extension of classical pumping, and discussed elsewhere on this forum.