The answer is apparently yes.
Recall the Grzegorczk hierarchy $\mathcal{E}^n$. This a hierarchy which classifies computable functions based on their growth rate, see Wikipedia: Grzegorczyk hierarchy.
Now on this hierarchy can be extended to ordinal level, see Wikipedia: Fast Growing Hierarchy.
Then we have $\mathcal{E}^\omega:= \bigcup_{n<\omega} \mathcal{E}^n$, which is the same as the set of all primitive recursive functions.
On the next level $\mathcal{E}^{\omega+1}$ one find then a variant of the Ackermann function.
Now Cleave, Rose formulate in $\mathcal{E}^n$ arithmetic an arithmetical systems $\mathcal{E}^n$ corresponding to the set of functions $\mathcal{E}^n$. Further, they show there that $\mathcal{E}^{n+1}$ proves the consistency of $\mathcal{E}^n$ for $n<\omega$.
In $\mathcal{E}^\alpha$ and transfinite induction Rose claims that this result is also true for $n$ being an infinite ordinal.
Thus arithmetic with a variant of the Ackermann function proves consistency of PRA (which is the same as consistency of $I\Sigma_1$).