What is a polite and friendly reply for Muslim ladies when a non-Mahram offers his hand for a handshake?

How can one avoid being called an extremist in such situations?

Sometimes Allah tests us with situations in which it is almost impossible not to be called an extremist. We shouldn't be afraid of such a label as every practising Muslim gets called an extremist at some point in their life or another.

According to the majority of scholars, touching a non-Mahram is prohibited and this includes shaking hands, as per the Hadith, “It is better for an iron rod to be driven into the head of a man, than for him to touch a woman who is not permissible for him.” (Mu'jam al Kabir)

This is something we need to be wise about, in how we explain it to others. Since in modern culture, not shaking hands is considered rude, it can be tricky to avoid this without offending someone.

One can avoid this by having your hands full, or just apologizing and explaining that your religion doesn’t allow it.

Allah knows best