We define a dynamical simple symmetric random walk in one dimension, and show that there almost surely exist exceptional times at which the walk tends to infinity. This is in contrast to the usual dynamical simple symmetric random walk in one dimension, for which such exceptional times are known not to exist. In fact we show that the set of exceptional times has Hausdorff dimension $1/2$ almost surely, and give bounds on the rate at which the walk diverges at such times. We also show noise sensitivity of the event that our random walk is positive after $n$ steps. In fact this event is maximally noise sensitive, in the sense that it is quantitatively noise sensitive for any sequence $\varepsilon_n$ such that $n\varepsilon_n\to\infty$. This is again in contrast to the usual random walk, for which the corresponding event is known to be noise stable.