Query on consequences of $\mathsf{P=BPP}$
The conclusion is that the permanent does not have polynomial-size (constant-free) arithmetic circuits or $\mathsf{NEXP} \not\subseteq \mathsf{P/poly}$. Permanent not having polynomial-size constant-free arithmetic circuits is not known to imply $\mathsf{NEXP} \not\subseteq \mathsf{P/poly}$.

However, there are a couple more recent similar results whose conclusion is not a disjunction, but simply a lower bound on some polynomial family in $\mathsf{NEXP}$:


Jansen and Santhanam showed that if PIT is in $\mathsf{NSUBEXP}$ then there is a family of polynomial whose integer evaluation problem is in $\mathsf{NEXP}$ but which do not have poly-size constant-free arithmetic circuits.
Carmosino, Impagliazzo, Kabanets, and Kolokolova show that if PIT over a fixed finite field is in $\mathsf{NSUBEXP}$ then there is a multilinear polynomial whose evaluation problem over that same finite field is in $\mathsf{NE}$ that does not have poly-size arithmetic circuits. (This paper also has many other results of interest, including true hardness-to-randomness equivalences, avoiding many of the usual caveats.)