Augustin Goderniaux, you are definitely NOT hallucinating. You are experiencing the so-called "intrusive R" phenomenon.
Intrusive "R" occurs when speakers want to avoid two vowel sounds occurring next to each other and
acts as a connector, between a word-final schwa /ə/ spelt with an < a > (e.g. China, Lisa, America) and a new word beginning with a vowel sound.
Ex: Lisa –r– Edwards.
In some ways it’s a strange phenomenon because native speakers often don’t realise that they’re doing it!
Note, that the intrusive "R" is not spelled.
Don't confuse it with a "linking R", which is spelled, like in: care about,