  We investigate the number of straight lines contained in a K3 quartic surface
\(X\) defined over an algebraically closed field of characteristic 3. We prove
that if \(X\) contains 112 lines, then \(X\) is projectively equivalent to the
Fermat quartic surface; otherwise, \(X\) contains at most 67 lines. We improve
this bound to 58 if \(X\) contains a star (ie four distinct lines intersecting
at a smooth point of \(X\)). Explicit equations of three 1-dimensional families
of smooth quartic surfaces with 58 lines, and of a quartic surface with 8
singular points and 48 lines are provided.
