  Let $\ell$ be a prime number and $q$ be a power of $\ell$. Given an odd prime
number $p$ and an imaginary quadratic extension $F$ of the rational function
field $\mathbb{F}_q(T)$, let $\lambda_p(F)$ denote the Iwasawa
$\lambda$-invariant of the constant $\mathbb{Z}_p$-extension of $F$. We show
that for any number $r>0$ and all large enough values of $q\not\equiv
1\mod{p}$, there is a positive proportion of imaginary quadratic fields
$F/\mathbb{F}_q(T)$ with the property that $\lambda_p(F)\geq r$. The main
result is proved as a consequence of recent unconditional theorems of
Ellenberg-Venkatesh-Westerland on the distribution of class groups of imaginary
quadratic function fields.
