The wandering subspace problem for an analytic norm-increasing
m
m
-isometry
T
T
on a Hilbert space
H
\mathcal {H}
asks whether every
T
T
-invariant subspace of
H
\mathcal {H}
can be generated by a wandering subspace. An affirmative solution to this problem for
m
=
1
m=1
is ascribed to Beurling-Lax-Halmos, while that for
m
=
2
m=2
is due to Richter. In this paper, we capitalize on the idea of weighted shift on a one-circuit directed graph to construct a family of analytic cyclic
3
3
-isometries which do not admit the wandering subspace property and which are norm-increasing on the orthogonal complement of a one-dimensional space. Further, on this one-dimensional space, their norms can be made arbitrarily close to
1
1
. We also show that if the wandering subspace property fails for an analytic norm-increasing
m
m
-isometry, then it fails miserably in the sense that the smallest
T
T
-invariant subspace generated by the wandering subspace is of infinite codimension.