  Recently, Arutyunov, Bassi and Lacroix have shown that 2D non-linear sigma
model with a deformed $T^{1,1}$ background is classically integrable
[arXiv:2010.05573 [hep-th]]. This background includes a Kalb-Ramond two-form
with a critical value. Then the sigma model has been conjectured to be
non-integrable when the two-form is off critical. We confirm this conjecure by
explicitly presenting classical chaos. With a winding string ansatz, the system
is reduced to a dynamical system described by a set of ordinary differential
equations. Then we find classical chaos, which indicates non-integrability, by
numerically computing Poincar\'{e} sections and Lyapunov spectra for some
initial conditions.
