  The study of the inner region of the Milky Way's bulge is hampered by high
interstellar extinction and extreme source crowding. Sensitive high angular
resolution near-infrared imaging is needed to study stellar populations in such
a complex environment. We use the 0.2$''$ angular resolution $JHK_s$ data from
the GALACTICNUCLEUS survey to study the stellar population within two
$8.0'\times 3.4'$ fields, about 0.6$^\circ$ and 0.4$^\circ$ to the Galactic
north of the Milky Way's centre and to compare it with one in the immediate
surroundings of Sagittarius A*. We also characterise the extinction curve of
the two fields. The average interstellar extinction to the outer and the inner
field is $A_{K_s} \sim 1.20 \pm 0.08$ mag and $\sim 1.48 \pm 0.10$ mag,
respectively. We present $K_{s}$ luminosity functions that are complete down to
at least 2 mag below the red clump (RC). We detect a feature in the luminosity
functions that is fainter than the RC by $0.80\pm0.03$ and $0.79\pm0.02$ mag,
respectively, in the $K_s$-band. It runs parallel to the reddening vector. We
identify the feature as the red giant branch bump. Fitting $\alpha$-enhanced
BaSTI luminosity functions to our data, we find that a single old stellar
population of $\sim12.8 \pm 0.6$ Gyr and $Z = 0.040 \pm 0.003$ provides the
best fit. We obtain that the stellar population in the innermost bulge is old,
similar to the one at larger distances from the Galactic plane, and that its
metallicity increases down to distances as short as about 60 pc from the centre
of the Milky Way. Comparing it with previous known values at larger latitudes
($|b|>2^\circ$), our results favour a flattening of the gradient at
$|b|<2^\circ$. As a secondary result we obtain that the extinction index in the
studied regions agrees the value of $\alpha = 2.30\pm0.08$, derived in
Nogueras-Lara et al. 2018 for the very Galactic centre.
