Is a flame made of plasma?

In normal, bonfire type situations, no.

Plasma flames for thermal spraying can produce temperatures around 7,000 to 20,000K far above the melting temperature (and vapour temperature) of any known material. This excedes most "normal" fires' temperatures.

http://www.gordonengland.co.uk/pft.htm copy

A flame is substance that is glowing and mitting light due to high temperature that changes only a small part of it into another type of substance called plasma. The higher the temperature the more is converted to plasma.

No. A flame is merely heated gases and glowing molecules.

Plasma is another state of matter. There is such a thing as a "plasma flame", but the word "flame" is used there only as a convenience. "Plasma jet" or "plasma flow" would be more accurate.

Your friend's assertion simply shows that he is not well informed. Most likely, it was a question framed as an assertion, with an implicit request for a contradiction that would inform him about the subject. Many people, particularly the uneducated, are embarrassed to admit what they don't know and feel humiliated if they have to ask a question, so they shoot their mouths off making absurd statements, just so they can get the correct facts.