The article deals with the \(2n\)th-order Sturm–Liouville difference operators \[ L(y)_k = \sum_{\nu=0}^n (-\Delta)^\nu (r_k^{[\nu]} \Delta^\nu y_{k-\nu}), \qquad k \in {\mathbb Z}, \] and the associated matrix operators \[ ({\mathcal T}y)_k = \sum_{j=k-n}^{k+n} t_{kj}y_j, \qquad k \in {\mathbb N}, \] defined by infinite symmetric banded matrices \(T = (t_{\mu\nu})\) with \(t_{\mu\nu} = t_{\nu\mu}\), \(\mu,\nu \in {\mathbb Z}\), \(t_{\mu\nu} = 0\) for \(|\mu - \nu| > n\). The arguments of the authors are based on the relationship between \(L\) and linear Hamiltonian difference systems \[ \Delta x_k = Ax_{k+1} + B_ku_k, \qquad \Delta u_k = C_kx_{k+1} - A^Tu_k \] whose recessive solutions allow them to introduce the class of \(p\)-critical Sturm–Liouville operators. The main result (Theorem 1) states that, among arbitrary small perturbations of a \(p\)-critical \(2n\)th-order Sturm–Liouville difference operator \(L\), there exist perturbations for which the perturbated operator \(\hat{L}\) is supercritical. In the end of the article the one-term difference operator \(L(y) = (-\Delta0^n(r_k\Delta^ny_k)\) is considered in details.