  We present an astrometric study of the proper motions (PMs) in the core of
the globular cluster NGC 6441. The core of this cluster has a high density and
observations with current instrumentation are very challenging. We combine
ground-based, high-angular-resolution NACO@VLT images with Hubble Space
Telescope ACS/HRC data and measure PMs with a temporal baseline of 15 yr for
about 1400 stars in the centermost 15 arcseconds of the cluster. We reach a PM
precision of $\sim$30 $\mu$as yr$^{-1}$ for bright, well-measured stars.
  Our results for the velocity dispersion are in good agreement with other
studies and extend already-existing analyses of the stellar kinematics of NGC
6441 to its centermost region never probed before. In the innermost arcsecond
of the cluster, we measure a velocity dispersion of (19.1 $\pm$ 2.0) km
s$^{-1}$ for evolved stars. Because of its high mass, NGC 6441 is a promising
candidate for harbouring an intermediate-mass black hole (IMBH). We combine our
measurements with additional data from the literature and compute dynamical
models of the cluster. We find an upper limit of $M_{\rm IMBH} < 1.32 \times
10^4\,\textrm{M}_\odot$ but we can neither confirm nor rule out its presence.
We also refine the dynamical distance of the cluster to $12.74^{+0.16}_{-0.15}$
kpc.
  Although the hunt for an IMBH in NGC 6441 is not yet concluded, our results
show how future observations with extremely-large telescopes will benefit from
the long temporal baseline offered by existing high-angular-resolution data.
