  In this paper, we study inhomogeneous Diophantine approximation over the
completion $K_v$ of a global function field $K$ (over a finite field) for a
discrete valuation $v$, with affine algebra $R_v$. We obtain an effective upper
bound for the Hausdorff dimension of the set \[
\mathbf{Bad}_A(\epsilon)=\left\{\boldsymbol{\theta}\in K_v^{\,m} :
\liminf_{(\mathbf{p},\mathbf{q})\in
  R_v^{\,m} \times R_v^{\,n}, \|\mathbf{q}\|\to \infty} \|\mathbf{q}\|^n
\|A\mathbf{q}-\boldsymbol{\theta}-\mathbf{p}\|^m \geq \epsilon \right\}, \] of
$\epsilon$-badly approximable targets $\boldsymbol{\theta}\in K_v^{\,m}$ for a
fixed matrix $A\in\mathscr{M}_{m,n}(K_v)$, using an effective version of
entropy rigidity in homogeneous dynamics for an appropriate diagonal action on
the space of $R_v$-grids. We further characterize matrices $A$ for which
$\mathbf{Bad}_A(\epsilon)$ has full Hausdorff dimension for some $\epsilon>0$
by a Diophantine condition of singularity on average. Our methods also work for
the approximation using weighted ultrametric distances.
