  We establish an optimal (topological) irreducibility criterion for $p$-adic
Banach principal series of $\mathrm{GL}_{n}(F)$, where $F/\mathbb{Q}_p$ is
finite and $n \le 3$. This is new for $n = 3$ as well as for $n = 2$, $F \ne
\mathbb{Q}_p$ and establishes a refined version of Schneider's conjecture
[Sch06, Conjecture 2.5] for these groups.
