Two-dimensional flat band on the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>UTe</mml:mi> <mml:mn>2</mml:mn> </mml:msub> <mml:mrow> <mml:mo>(</mml:mo> <mml:mn>011</mml:mn> <mml:mo>)</mml:mo> </mml:mrow> </m

Scanning tunneling microscopy (STM) measurements have been extensively performed on the easily cleavable (011) surface of UTe$_2$, using both normal-metal and superconducting tips. Motivated by these experiments, we theoretically investigate the topological surface states on the (011) surface of UTe$_2$. We find that a two-dimensional nearly flat band emerges in the $B_{3u}$ state, giving rise to a pronounced zero-energy peak in the surface density of states. This flat band is supported by two key mechanisms: (i)~nontrivial Berry phases defined at multiple momenta give rise to low-energy in-gap states, and (ii)~weak spin conservation allows the gap function to acquire phase winding. Furthermore, to investigate the relation between the zero-bias peak observed in recent STM experiments with a superconducting tip and the topological surface states, we calculate the nonequilibrium dc tunneling current in a junction between an $s$-wave superconductor and the (011) surface of UTe$_2$. Our results provide crucial insights into the superconducting pairing symmetry realized in UTe$_2$.