  We prove that if an initial datum to the incompressible Navier-Stokes
equations in any critical Besov space $\dot B^{-1+\frac
3p}_{p,q}(\mathbb{R}^3)$, with $3 <p,q< \infty$, gives rise to a strong
solution with a singularity at a finite time $T>0$, then the norm of the
solution in that Besov space becomes unbounded at time $T$. This result, which
treats all critical Besov spaces where local existence is known, generalizes
the result of Escauriaza, Seregin and Sverak (Uspekhi Mat. Nauk
58(2(350)):3-44, 2003) concerning suitable weak solutions blowing up in
$L^3(\mathbb{R}^3)$. Our proof uses profile decompositions and is based on our
previous work (Math. Ann. 355(4):1527--1559, 2013) which provided an
alternative proof of the $L^3(\mathbb{R}^3)$ result. For very large values of
$p$, an iterative method, which may be of independent interest, enables us to
use some techniques from the $L^3(\mathbb{R}^3)$ setting.
  (To appear in Communications in Mathematical Physics.)
