I will work over $\mathbb C$. Although I have not checked, the example below should work for characteristic different from $3$.
To exhibit a degree $d$ projective surface $S \subset \mathbb P^3$ not containing any line you can consider surfaces of the form $t^d = f(x,y,z)$ where $f$ is homogeneous polynomial of degree $d$.
Let $C \subset \mathbb P^2$ be the curve determined by the polynomial $f$ and $\pi: S \to \mathbb P^2$ be a projection from a point outside $p$. If $\ell$ is a line contained in $S$ then $\pi(\ell)$ is a line tangent to $C$ at a total inflection point $q$, i.e. the contact between $C$ and the line $\pi(\ell)$ at $q$ is of order $d$. For details see Section 6 Counting lines on surfaces by Boissière and Sarti.
This reduces the problem of finding a surface without lines to the one of finding an algebraic curve without total inflection points. For quartic curves we have not to look much, as Klein determined all the inflections of his famous quartic $$ x^3y + y^3 z+ z^3 x = 0. $$ All its $24$ inflection points are simple. See for instance Jeremy Gray's paper in the book The Eightfold Way: The Beauty of Klein's Quartic Curve. Thus the surface $$ t^4 - x^3 y - y^3 z - z^3 x =0 $$ has no invariant lines.