  We investigate continuously self-similar solutions of four-dimensional
Einstein-Maxwell-dilaton theory supported by charged null fluids. We work under
the assumption of spherical symmetry and the dilaton coupling parameter $a$ is
allowed to be arbitrary. First, it is proved that the only such vacuum
solutions with a time-independent asymptotic value of the dilaton necessarily
have vanishing electric field, and thus reduce to Roberts' solution of the
Einstein-dilaton system. Allowing for additional sources, we then obtain
Vaidya-like families of self-similar solutions supported by charged null
fluids. By continuously matching these solutions to flat spacetime along a null
hypersurface one can study gravitational collapse analytically. Capitalizing on
this idea, we compute the critical exponent defining the power-law behavior of
the mass contained within the apparent horizon near the threshold of black hole
formation. For the heterotic dilaton coupling $a=1$ the critical exponent takes
the value $1/2$ typically observed in similar analytic studies, but more
generally it is given by $\gamma=a^2(1+a^2)^{-1}$. The analysis is complemented
by an assessment of the classical energy conditions. Finally, and on a
different note, we report on a novel dyonic black hole spacetime, which is a
time-dependent vacuum solution of this theory. In this case, the presence of
constant electric and magnetic charges naturally breaks self-similarity.
