A first-order relativistic wave equation is constructed in five dimensions. Its solutions are eight-component spinors, interpreted as single-particle fermion wave functions in four-dimensional space-time. Use of a "cylinder condition" (the removal of explicit dependence on the fifth coordinate) reduces each eight-component solution to a pair of degenerate four-component spinors. It is shown that, when the cylinder condition is applied, the results obtained from the new equation are the same as those obtained from the Dirac equation. Without the cylinder condition, on the other hand, the equation implies the existence of a scalar potential, and for zero-mass particles it leads to a four-dimensional fermionic equation analogous to Maxwell's equation with sources.