  In this paper, we are interested in studying when an element $z$ in the
projective tensor product $X \widehat{\otimes}_\pi Y$ turns out to be a
Daugavet point. We prove first that, under some hypothesis, the assumption of
$X \widehat{\otimes}_\pi Y$ having the Daugavet property implies the existence
of a great amount of isometries from $Y$ into $X^*$. Having this in mind, we
provide methods for constructing non-trivial Daugavet points in $X
\widehat{\otimes}_\pi Y$. We show that $C(K)$-spaces are examples of Banach
spaces such that the set of the Daugavet points in $C(K) \widehat{\otimes}_\pi
Y$ is weakly dense when $Y$ is a subspace of $C(K)^*$. Finally, we present some
natural results on when an elementary tensor $x \otimes y$ is a Daugavet point.
