Topologically massive $p$-form gauge fields in arbitrary dimensions

$newcommand{d}{mathrm{d}}$In $d=2p+1$ dimensions one can have topologically massive $p$-form abelian gauge fields $AinOmega^p(X_{2p+1})$ by considering a Maxwell–Chern–Simons action:
$$S[A] = int_{X_{2p+1}} frac{1}{2g^2}d Awedgestard A + ifrac{k}{2}Awedged A. $$
In this case, it is easy to show that the gauge field has mass $g^2 |k|$.

In dimensions $dneq 2p+1$ is there a similar way to gap $A$ out?

Of course, a Chern–Simons interaction does not make sense, but I was thinking something like a BF interaction, with appropriate form-degree fields. What is the correct treatment?

Addendum: Indeed, if we consider an action as
$$ S[A,B] = int_{X_d} frac{1}{2}d Awedgestard A+im Bwedge d A + frac{1}{2}d Bwedgestard B,$$
where $A$ again $inOmega^{p}(X_d)$ and $BinOmega^{d-p-1}(X_d)$, we do get $A$ to be gapped, but at the cost of an extra gapped $B$. Can we somehow kill $B$? Also this works only whenever $dgeqslant p+1$. What about $d<p+1$?