  On 2019 October 25, the Fermi-Large Area Telescope observed the first
gamma-ray flare from the radio-loud narrow-line Seyfert 1 (NLSy 1) galaxy PKS
2004$-$447 ($z=0.24$). We report on follow-up observations in the radio,
optical-UV, and X-ray bands that were performed by ATCA, the Neil Gehrels Swift
observatory, XMM-Newton, and NuSTAR, respectively, and our multi-wavelength
analysis. We study the variability across all energy bands and additionally
produce $\gamma$-ray light curves with different time binnings to study the
variability on short timescales during the flare. We examine the X-ray spectrum
from 0.5$-$50 keV by describing the spectral shape with an absorbed power law.
We analyse multi-wavelength datasets before, during, and after the flare and
compare these with a low activity state of the source by modelling the
respective SEDs with a one-zone synchrotron inverse Compton radiative model.
Finally, we compare our results to gamma-ray flares previously observed from
other $\gamma$-loud NLSy 1 galaxies. At gamma-ray energies (0.1$-$300 GeV) the
flare reached a total maximum flux of
$(2.7\pm0.6)\times10^{-6}$~ph~cm$^{-2}$~s$^{-1}$ in 3-hour binning. With a
photon index of $\Gamma_{0.1-300\mathrm{GeV}}=2.42\pm0.09$ during the flare,
this corresponds to an isotropic gamma-ray luminosity of
$(2.9\pm0.8)\times10^{47}\,\mathrm{erg}\,\mathrm{s}^{-1}$. The $\gamma$-ray,
X-ray, and optical-UV light curves covering the end of September to the middle
of November show significant variability, and we find indications for
flux-doubling times of $\sim 2.2$~hours at $\gamma$-ray energies. During the
flare, the SED exhibits large Compton dominance. While the increase in the
optical-UV range can be explained by enhanced synchrotron emission, the
elevated $\gamma$-ray flux can be accounted for by an increase in the bulk
Lorentz factor of the jet, similarly observed for flaring gamma-ray blazars.
