  We studied the radio source associated with the ultraluminous X-ray source in
NGC 5408 (L_X ~ 10^{40} erg/s). The radio spectrum is steep (index ~ -1),
consistent with optically-thin synchrotron emission, not with flat-spectrum
core emission. Its flux density (~ 0.28 mJy at 4.8 GHz, at a distance of 4.8
Mpc) was the same in the March 2000 and December 2004 observations, suggesting
steady emission rather than a transient outburst. However, it is orders of
magnitude higher than expected from steady jets in stellar-mass microquasar.
Based on its radio flux and spectral index, we suggest that the radio source is
either an unusually bright supernova remnant, or, more likely, a radio lobe
powered by a jet from the black hole. Moreover, there is speculative evidence
that the source is marginally resolved with a radius ~ 30 pc. A faint HII
region of similar size appears to coincide with the radio and X-ray sources,
but its ionization mechanism remains unclear. Using a self-similar solution for
the expansion of a jet-powered electron-positron plasma bubble, in the
minimum-energy approximation, we show that the observed flux and (speculative)
size are consistent with an average jet power ~ 7 x 10^{38} erg/s ~ 0.1 L_X ~
0.1 L_{Edd}, an age ~ 10^5 yr, a current velocity of expansion ~ 80 km/s. We
briefly discuss the importance of this source as a key to understand the
balance between luminosity and jet power in accreting black holes.
