Can a unit delay in discrete time be represented by exponential functions?
In continuous time, we can, though it is a mathematical curiosity of little practical use. Suppose we have a smooth continuous-time signal $x$. Using the Taylor expansion,

$$x (t-1) = x(t) - \dot x (t) + \frac 12 \ddot x (t) - \cdots = \left( 1 - \frac{\mathrm d}{\mathrm d t} + \frac 12 \left(\frac{\mathrm d}{\mathrm d t}\right)^2 - \cdots \right) x(t) = \color{blue}{\exp \left( - \frac{\mathrm d}{\mathrm d t} \right) \, x (t)}$$