Can the order of 2 modulo $p$ be arbitrarily small (relative to $p-1$) if p is a Wieferich prime

Question

Let $k={\rm ord}_p\ 2$ be the multiplicative order of 2 modulo p. Can the ratio $\frac{p-1}{k}$ be arbitrarily large if $p$ is a Wieferich prime? This is known to be true without the Wieferich restriction (related post) using Chebotarev's density theorem, but what happens if you introduce the restriction that $p$ is a Wieferich prime? Of greater interest to me is to know whether $p=O(k^t)$ for some fixed positive integer $t$ or not.