I stumbled upon these very nice looking notes by Brian Lawrence (https://math.uchicago.edu/~brianrl/notes/fibonacci.pdf) on the period of Fibonacci numbers over finite fields. In them, he shows that the period of the Fibonacci sequence over $\mathbb{F}_p$ divides $p$ or $p-1$ or $p+1$.

I am wondering if there are explicit lower bounds on this period. Is it true, for instance, that as $p \to \infty$, so does the order?

A quick calculation on my computer shows that there are some "large" primes with period under 100.

9901 66
19489 58
28657 92