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 $(2.7pm0.6)times10^{-6}$~ph~cm$^{-2}$~s$^{-1}$ in 3-hour binning. With a photon index of $Gamma_{0.1-300mathrm{GeV}}=2.42pm0.09$ during the flare, this corresponds to an isotropic gamma-ray luminosity of $(2.9pm0.8)times10^{47},mathrm{erg},mathrm{s}^{-1}$. 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.
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