  The small binding energy of the hypertrition leads to predictions of
non-existence of bound hypernuclei for isotriplet three-body systems such as
$nn\Lambda$. However, invariant mass spectroscopy at GSI has reported events
that may be interpreted as the bound $nn\Lambda$ state. The $nn\Lambda$ state
was sought by missing-mass spectroscopy via the $(e,e'K^+)$ reaction at
Jefferson Lab's experimental Hall A. The present experiment has higher
sensitivity to the $nn\Lambda$-state investigation in terms of better precision
by a factor of about three. The analysis shown in this article focuses on the
derivation of the reaction cross-section for the
$^3{\rm{H}}(\gamma^{*},K^+)\textrm{X}$ reaction. Events that were detected in
an acceptance, where a Monte Carlo simulation could reproduce the data well
($|\delta p/p| < 4\%$), were analyzed to minimize the systematic uncertainty.
No significant structures were observed with the acceptance cuts, and the upper
limits of the production cross-section of the $nn\Lambda$ state were obtained
to be $21$ and $31~\rm{nb/sr}$ at the $90\%$ confidence level when theoretical
predictions of $(-B_{\Lambda}, \Gamma) = (0.25,0.8)$ and $(0.55, 4.7)$ MeV,
respectively, were assumed. The cross-section result provides valuable
information for examining the existence of $nn\Lambda$.
