Biblical basis for "We are the living law"?

Your nemesis(!) is at least partially right. Doctrinally, at least from an Evangelical perspective, he or she is spot on. The person's behavior, on the other hand, is far from Christ-like.

Paul is quite clear in Romans that the Law of God is a good thing (as Martha Stewart is wont to say--except she leaves out the words about God's Law):

So then, the Law is holy, and the commandment is holy and righteous and good (7:12).

He also is clear about the role God's Law is to play in the lives of believers.

For the law of the Spirit of life in Christ Jesus has set you free from the law of sin and of death. For what the Law could not do, weak as it was through the flesh, God did: sending His own Son in the likeness of sinful flesh and as an offering for sin. He condemned sin in the flesh, so that the requirement of the Law might be fulfilled in us, who do not walk according to the flesh but according to the Spirit
(Romans 8:2-4, NASB, my italics)

Put differently, this time in Jeremiah's words,

"The days are coming," declares the LORD, "when I will plant the house of Israel and the house of Judah with the offspring of men and of animals. Just as I watched over them to uproot and tear down, and to overthrow, destroy and bring disaster, so I will watch over them to build and to plant," declares the LORD. "In those days people will no longer say, 'The fathers have eaten sour grapes, and the children's teeth are set on edge.' Instead, everyone will die for his own sin; whoever eats sour grapes--his own teeth will be set on edge. "The time is coming," declares the LORD, "when I will make a new covenant with the house of Israel and with the house of Judah. It will not be like the covenant I made with their forefathers when I took them by the hand to lead them out of Egypt, because they broke my covenant, though I was a husband to them, " declares the LORD. "This is the covenant I will make with the house of Israel after that time," declares the LORD. "I will put my law in their minds and write it on their hearts. I will be their God, and they will be my people"
(31:27-33 my italics)

And in Ezekiel's words,

"For I will take you out of the nations; I will gather you from all the countries and bring you back into your own land. I will sprinkle clean water on you, and you will be clean; I will cleanse you from all your impurities and from all your idols. I will give you a new heart and put a new spirit in you; I will remove from you your heart of stone and give you a heart of flesh. And I will put my Spirit in you and move you to follow my decrees and be careful to keep my laws"
(36:26 ff., NIV, my italics)

In this Age of Grace in which we live, the Holy Spirit empowers us to fulfill the Law's just requirements. We are not to be antinomianists (i.e., people who consider themselves free from the Law of God, since we're "saved by Grace"), having the attitude, "Hey, the more I sin, the more gracious God will be!" To that statement, Paul would lend us his emphatic, "By no means!" or "God forbid!" (see Romans 6:1 ff.).

The point Paul and others make in this regard is this: While we cannot earn our salvation by fulfilling the Law of God--for then, our salvation would be of works and not of faith, we have been freed from the curse attached to our disobedience to the Law.

"Behold, all souls are mine; as the soul of the father, so also the soul of the son is mine: the soul that sinneth, it shall die.
(Ezekiel 18:4)

We have been freed from fulfilling the just requirements of the Law for our salvation. Jesus made this freedom possible through his shed blood. On the other hand, we have not been freed from fulfilling the just requirements of the Law in the power of the Holy Spirit, who has been given to us once we are regenerated. The former has to do with salvation. The latter has to do with sanctification.

In other words, we cannot be saved by obeying the Law, because whoever has committed one sin (or a million sins)--and we all have--"shall die." We can, however, prove to ourselves and others that we are indeed saved by fulfilling the Law's just demands through our way of life as followers of Jesus Christ. For Christ has set us free for this very purpose, since He fulfilled every single one of the Law's just requirements as the sinless and spotless Lamb of God. That is why he could be the perfect sacrifice for sin.

In conclusion, your "friend" is correct, doctrinally. In his obligation to live a life of love, he is sadly lacking. Go to him or her in private, in keeping with Matthew 5:23-24 and Matthew 18:15-18. Best wishes to you!