  We study a fractional version of the two-dimensional discrete nonlinear
Schr\"{o}dinger (DNLS) equation, where the usual discrete Laplacian is replaced
by its fractional form that depends on a fractional exponent $s$ that
interpolates between the case of an identity operator ($s=0$) and that of the
usual discrete 2D Laplacian ($s=1$). This replacement leads to a long-range
coupling among sites that, at low values of $s$, decreases the bandwidth and
leads to quasi-degenerate states. The mean square displacement of an
initially-localized excitation is shown to be ballistic at all times with a
`speed' that increases monotonically with the fractional exponent $s$. We also
compute the nonlinear modes and their stability for both, bulk and surface
modes. The modulational stability is seen to increase with an increase in the
fractional exponent. The trapping of an initially localized excitation shows a
selftrapping transition as a function of nonlinearity strength, whose threshold
increases with the value of $s$. In the linear limit, there persists a linear
trapping at small $s$ values. This behavior is connected with the decrease of
the bandwidth and its associated increase in quasi-degeneracy.
