My question is, if a non-Jewish man marries a Jewish woman, since the marriage is invalid, are the children Jewish? Specifically, is there any source in the Mishnah which says that they are not Jewish? Are there opinions either way?
This is pretty much open-and-shut.
Mishna, Kiddushin 3:13.
So the mishna is spelling out that if mom is non-Jewish, the child is not Jewish. It's implying the converse fairly strongly, that if mom's Jewish so is the child; but the Gemara does quote some different opinions about it. (Rabbi Akiva, fascinatingly, is of the opinion that if dad's not Jewish but mom is, the child is a mamzer. We don't follow his opinion.)
Rambam rules (Prohibited Relations 15:4):
So in the case of your question, it's actually precisely because no marriage is valid that we ignore the non-Jewish father completely -- leaving the child Jewish.
For theory's sake, long ago there had been an opinion that both parents must be Jewish for the child to be Jewish -- we don't follow that opinion. But Rabbi Aharon Rakeffet-Rothkoff tells the story of a female student of his whose dad was not Jewish who was constantly upset about the possibility of this minority opinion. Telling her "we don't follow that opinion" wasn't calming her fears. So he suggested that when she gets married, she'll be going to the mikvah anyhow, so she could ask a panel of three rabbis to witness the dunking and call it a conversion, just in case. Halachically it's completely not necessary, but it made this woman feel a lot better by getting rid of her doubts.