  We explore the discovery prospect of the doubly-charged component of an
$SU(2)_L$-triplet scalar at the future $e^- p$ collider FCC-eh, proposed to
operate with an electron beam energy of 60 GeV and a proton beam energy of 50
TeV. We consider the associated production of the doubly-charged Higgs boson
along with leptons and jet(s), and its subsequent prompt decay to same-sign
lepton pair. This occurs for ${\cal O}(1)$ Yukawa coupling of the scalar
triplet with charged leptons, which is expected for reasonably small vacuum
expectation values of the neutral component of the triplet field that governs
the neutrino mass generation in the type-II seesaw. We present our analysis for
two different final states, $3l+\ge1j$ and an inclusive $\ge2l+\ge1j$ channel.
Considering its decay to electrons only, we find that the doubly-charged Higgs
with mass around a TeV could be observed at $3\sigma$ confidence level with
$\mathcal{O}(200)\, \rm{fb}^{-1}$ of integrated luminosity, while masses up to
2 TeV could be probed within a few years of data accumulation. The signal
proposed here becomes essentially background-free, if it is triggered in the
$\mu\mu$ mode and a $5\sigma$ discovery is achievable in this channel for a
TeV-scale doubly-charged Higgs with an integrated luminosity as low as
$\mathcal{O} (50)\, \rm{fb}^{-1}$. We also highlight the sensitivity of FCC-eh
to the Yukawa coupling responsible for production of the doubly-charged Higgs
boson as a function of its mass in both the $ee$ and $\mu\mu$ channels.
