  We investigate the effect of introducing nearest-neighbor $p$-wave
superconducting pairing to both the static and kicked extended Harper model
with two periodic phase parameters acting as artificial dimensions to simulate
three-dimensional systems. It is found that in both the static model and the
kicked model, by varying the $p$-wave pairing order parameter, the system can
switch between a fully gapped phase and a gapless phase with point nodes or
line nodes. The topological property of both the static and kicked model is
revealed by calculating corresponding topological invariants defined in the
one-dimensional lattice dimension. Under open boundary conditions along the
physical dimension, Majorana flat bands at energy zero (quasienergy zero and
$\pi$) emerge in the static (kicked) model at the two-dimensional surface
Brillouin zone. For certain values of pairing order parameter, (Floquet)
Su-Schrieffer-Heeger-like edge modes appear in the form of arcs connecting
different (Floquet) Majorana flat bands. Finally, we find that in the kicked
model, it is possible to generate two controllable Floquet Majorana modes, one
at quasienergy zero and the other at quasienergy $\pi$, at the same parameter
values.
