  (Abridged) We investigate the Lyman $\alpha$ emitter luminosity function (LAE
LF) within the redshift range $2.9 \leq z \leq 6$ from the first instalment of
the blind integral field spectroscopic survey MUSE-Wide. This initial part of
the survey probes a region of 22.2 arcmin$^2$ in the CANDELS/GOODS-S field. The
dataset provided us with 237 LAEs from which we construct the LAE LF in the
luminosity range $42.2 \leq \log L_\mathrm{Ly\alpha} [\mathrm{erg\,s}^{-1}]\leq
43.5$ within a volume of $2.3\times10^5$ Mpc$^3$. For the LF construction we
utilise three different non-parametric estimators: The classical
$1/V_\mathrm{max}$ method, the $C^{-}$ method, and an improved binned estimator
for the differential LF. All three methods deliver consistent results, with the
cumulative LAE LF being $\Phi(\log L_\mathrm{Ly\alpha} [\mathrm{erg\,s}^{-1}] =
43.5) \simeq 3\times 10^{-6}$ Mpc$^{-3}$ and $\Phi(\log L_\mathrm{Ly\alpha}
[\mathrm{erg\,s}^{-1}] = 42.2) \simeq 2 \times 10^{-3}$ Mpc$^{-3}$ towards the
bright- and faint-end of our survey, respectively. By employing a
non-parametric statistical test, as well as by comparing the full sample to
sub-samples in redshift bins, we find no supporting evidence for an evolving
LAE LF over the probed redshift and luminosity range. We determine the
best-fitting Schechter function parameters $\alpha = -1.84^{+0.42}_{-0.41}$ and
$\log L^* [\mathrm{erg\,s}^{-1}] = 42.2^{+0.22}_{-0.16}$ with the corresponding
normalisation $\log \phi^* [\mathrm{Mpc}^{-3}] = -2.71$. When correcting for
completeness in the LAE LF determinations, we take into account that LAEs
exhibit diffuse extended low surface-brightness haloes. We compare the
resulting LF to one obtained where we apply a correction assuming compact
point-like emission. We find that the standard correction underestimates the
LAE LF at the faint end of our survey by a factor of 2.5.
