  Identifying very high-redshift galaxies is crucial for understanding the
formation and evolution of galaxies. However, many questions still remain, and
the uncertainty on the epoch of reionization is large. In this approach, some
models allow a double-reionization scenario, although the number of confirmed
detections at very high $z$ is still too low to serve as observational proof.
The main goal of this project is studying whether we can search for
Lyman-$\alpha$ emitters (LAEs) at $z \sim 9$ using a narrow-band (NB) filter
that was specifically designed by our team and was built for this experiment.
We used the NB technique to select candidates by measuring the flux excess due
to the Ly$\alpha$ emission. The observations were taken with an NB filter (full
width at half minimum of 11 nm and central wavelength $\lambda_{c} = 1.257
\mu$m) and the CIRCE near-infrared camera for the GTC. We describe a data
reduction procedure that was especially optimized to minimize instrumental
effects. With a total exposure time of 18.3 hours, the final NB image covers an
area of $\sim 6.7$ arcmin$^{2}$, which corresponds to a comoving volume of $1.1
\times 10^{3}$ Mpc$^{3}$ at $z = 9.3$. We pushed the source detection to its
limit, which allows us to analyze an initial sample of 97 objects. We detail
the different criteria we applied to select the candidates. The criteria
included visual verifications in different photometric bands. None of the
objects resembled a reliable LAE, however, and we found no robust candidate
down to an emission-line flux of $2.9 \times 10^{-16}$ erg s$^{-1} $cm$^{-2}$,
which corresponds to a Ly$\alpha$ luminosity limit of $3 \times 10^{44}$ erg
s$^{-1}$. We derive an upper limit on the Ly$\alpha$ luminosity function at $z
\sim 9$ that agrees well with previous constraints. We conclude that deeper and
wider surveys are needed to study the LAE population at the cosmic dawn.
