We investigate the Lyman alpha emitter (LAE) luminosity function (LF) within the redshift range 2.9 <= z <= 6 from the first instalment of the blind integral field spectroscopic MUSE-Wide survey. This initial part of the survey probes a region of 22.2 arcmin(2) in the CANDELS/GOODS-S field (24 MUSE pointings with 1h integrations). The dataset provided us with 237 LAEs from which we construct the LAE LF in the luminosity range 42.2 <= log L-Ly alpha[erg s(-1)] <= 43.5 within a volume of 2.3 x 10(5) Mpc(3). For the LF construction we utilise three different non-parametric estimators: the classical 1/V-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-Ly alpha[erg s(-1)] = 43.5) similar or equal to 3 x 10(-6) Mpc(-3) and Phi(log L-Ly alpha[erg s(-1)] = 42.2) similar or equal to 2 x 10(-3) Mpc(-3) towards the bright and faint end of our survey, respectively. By employing a non-parametric statistical test, and by comparing the full sample to subsamples in redshift bins, we find no supporting evidence for an evolving LAE LF over the probed redshift and luminosity range. Using a parametric maximum-likelihood technique we determine the best-fitting Schechter function parameters alpha = -1.84(-0.41)(+0.42) and log L*[erg s(-1)] = 42.2(-0.16)(+0.22) with the corresponding normalisation log phi*[Mpc(-3)] = -2.71. However, the dynamic range in Ly alpha luminosities probed by MUSE-Wide leads to a strong degeneracy between alpha and L*. Moreover, we find that a power-law parameterisation of the LF appears to be less consistent with the data compared to the Schechter function, even so when not excluding the X-Ray identified AGN from the sample. When correcting for completeness in the LAE LF determinations, we take into account that LAEs exhibit diffuse extended low surface brightness halos. We compare the resulting LF to one obtained by applying 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. Contrasting our results to the literature we find that at log L-Ly alpha[erg s(-1)] less than or similar to 42.5 previous LAE LF determinations from narrow-band surveys appear to be affected by a similar bias.