  We prove linear independence of indefinite iterated Eisenstein integrals over
the fraction field of the ring of formal power series $\mathbb{Z}[[q]]$. Our
proof relies on a general criterium for linear independence of iterated
integrals, which has been established by Deneufch{\^a}tel, Duchamp, Minh and
Solomon. As a corollary, we obtain $\mathbb{C}$-linear independence of
indefinite iterated Eisenstein integrals, which has applications to the study
of elliptic multiple zeta values, as defined by Enriquez.
