= Solution
Suppose an <affine line> lies in $W$. Parametrize it as
$$
(x,y,z)=(a_1,a_2,a_3)+t(b_1,b_2,b_3).
$$
The defining equation would give the polynomial identity
$$
(a_1+tb_1)(a_2+tb_2)(a_3+tb_3)=1
\qquad\text{in }k[t].
$$
The only <unit in a polynomial ring>[units] of $k[t]$ are nonzero constants. Since a product is a unit only when every factor is a unit, all three linear polynomials must be constant, so $b_1=b_2=b_3=0$. The alleged line is a point, a contradiction. Therefore
$$
\boxed{W=V(xyz-1)\text{ contains no affine lines}}.
$$
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