Actually ... yes! Even when stress is applied below the yield stress, over time it can cause solids to flow. In industrial equipment, this process, called creep, significantly impacts the performance and lifetime of equipment. The flow of glaciers under the stresses created by their own weight is also a form of creep.
During a majority of the lifetime of the stressed solid, creep is described by this equation taken from the referenced WP articke:
$\dfrac{d\epsilon}{dt}=\dfrac{C\sigma^m}{d^b}\exp(\dfrac{-Q}{kT})$
This says that creep rate is increased with increasing stress $\sigma$, with finer grain size $d$ (because more grain boundaries make it easier for atoms to slide by each other, even as fine grain size is often desirable for other properties), and especially with temperature as shown by the exponential term (analogous to the activation energy in chemical reaction rates, not surprising since what happens is atoms are breaking/making bonds and moving). That temperature dependence means, in practice, creep will be extremely slow unless we are fairly close to the melting point of the solid, which in our ordinary experience isn't the case -- except, as alluded above, with high-temperature industrial equipment and with large, heavy ice formations.
Outside of such exceptions, we may have to wait longer than the current age of the Universe to see much of anything. Yet, given appropriate time and conditions, solids flow.
