Rabbinic Judaism assumes there is always a human biological father.
Again, the specific legal term mamzer doesn't mean "out of wedlock." It means "the product of two Jews whose union was adulterous or incestuous." (For certain categories of "incestuous.")
If a married pregnant becomes pregnant while her husband's been away at sea for over a year, we assume that there is a biological father who is not her husband.
(We give it up to one year; there are bell curves that show some fraction of a percentage of pregnancies [not necessarily viable ones] that can go up to that long.)
Absent any other information, though, the child would not have the specific laws of a mamzer: if a Jewish married woman has an affair with a non-Jewish man, the resulting child does not have the rules of a mamzer. That penalty was instituted specifically for the case where both biological parents were Jewish. In today's world it's more often likely the biological father is a non-Jew.
There's also discussion about the possibility of artificial insemination (the Talmud talks about the theoretical case where there was sperm in the bath water) or in-vitro fertilization. 20th Century rabbinic authorities debated whether a mamzer is generated by this scenario, if the mother was married and the sperm donor is a Jewish man not married to her. If a single woman gets artificially inseminated it's not a product of "adultery", so the child is again not a mamzer per se.
Absent any other information, we don't need to go there because maybe the biological father isn't Jewish.
Contemporary Jewish medical ethicists have discussed if we can prove via DNA that someone's biological parentage is a married Jewish woman and a specific Jewish man who wasn't her husband, would that give the child the status of mamzer or would we still allow for the possibility of artificial insemination / IVF. (Of course if we knew she'd undergone fertility treatments, a lab mix-up is certainly plausible.)