  We investigate the biased quenched trap model on top of a two-dimensional
lattice in the case of diverging expected dwell times. By utilizing the
double-subordination approach and calculating the return probability in $2$d,
we explicitly obtain the disorder averaged probability density function of the
particle's position as a function of time (for any given bias) in the limit of
large times ($t \rightarrow \infty$). The first and second moments are
calculated, and a formula for a general $\mu$-th moment is found. The behavior
of the first moment, i.e. $\langle x(t)\rangle$, presents non-linear response
both in time and in the applied external force $F_0$. While the non-linearity
in time occurs for any measurement time $t$, the non-linearity in $F_0$ is
expected only when $t\gtrsim \big(F_0 \left| \ln (F_0)\right|\big)^{-2 /
\alpha}$ where $\alpha=T/T_g$, for temperatures $T<T_g$. We support our
analytic results by comparison to numerical simulations.
