If a Jewess married to a Jew has a child from a non-Jew is that child a mamzer?
Is there agreement amongst the poskim about this?
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I don't know about modern poskim, but the Shulchan Aruch (Even HaEzer 4:19) rules that the child of a gentile and a Jewess independent of her marital status is not a mamzer but is disqualified from marrying a kohein (kohanim in general have stricter strictures restricting marriage). None of the standard commentaries there seem to disagree.
In addition to what @Double AA wrote in their answer, that is absolutely the view of modern poskim as well. Rav Ephraim Oshry has a thorough and moving teshuvah about it Shut Mimaamakim 2:24.
As an aside, the subsequent teshuvah (2:25) deals with whether such a person requires a conversion. I was surprised when I first read it that the question was not 100% black and white and there are opinions that such a person is not automatically Jewish. Rav Oshry concludes the overwhelming majority and correct approach is that such a person does not require conversion but it would be prudent to do one "just in case" and to avoid all doubt. (As far as I am aware, most rabbis today do not even do a "just in case" conversion, and I'm pretty familiar with inyanei gerut and "half Jews." Perhaps this is a result of greater community acceptance and/or emotional sensitivities of the people involved.).