  We perform a systematic Bayesian analysis of rotation vs. dispersion support
($v_{\rm rot} / \sigma$) in $40$ dwarf galaxies throughout the Local Volume
(LV) over a stellar mass range $10^{3.5} M_{\rm \odot} < M_{\star} < 10^8
M_{\rm \odot}$. We find that the stars in $\sim 80\%$ of the LV dwarf galaxies
studied -- both satellites and isolated systems -- are dispersion-supported. In
particular, we show that $6/10$ *isolated* dwarfs in our sample have $v_{\rm
rot} / \sigma < 1.0$. All have $v_{\rm rot} / \sigma \lesssim 2.0$. These
results challenge the traditional view that the stars in gas-rich dwarf
irregulars (dIrrs) are distributed in cold, rotationally-supported stellar
disks, while gas-poor dwarf spheroidals (dSphs) are kinematically distinct in
having dispersion-supported stars. We see no clear trend between $v_{\rm rot} /
\sigma$ and distance to the closest $\rm L_{\star}$ galaxy, nor between $v_{\rm
rot} / \sigma$ and $M_{\star}$ within our mass range. We apply the same
Bayesian analysis to four FIRE hydrodynamic zoom-in simulations of isolated
dwarf galaxies ($10^9 M_{\odot} < M_{\rm vir} < 10^{10} M_{\rm \odot}$) and
show that the simulated *isolated* dIrr galaxies have stellar ellipticities and
stellar $v_{\rm rot} / \sigma$ ratios that are consistent with the observed
population of dIrrs *and* dSphs without the need to subject these dwarfs to any
external perturbations or tidal forces. We posit that most dwarf galaxies form
as puffy, dispersion-dominated systems, rather than cold, angular
momentum-supported disks. If this is the case, then transforming a dIrr into a
dSph may require little more than removing its gas.
