I study the properties of the ring $R$ in the following example, but I don't know if $R$ is semisimple or not.
let $D$ be a field, $U$ an infinite dimension vector space over $D$, set $T=End(U)$, $K=soc(T)$ and let $R$ be the subring of $T$ generated by $K$ and the scalar transformation $1d$, for $d \in D$ .
This example was presented for the first time by C. Faith in Lectures on Injective Modules and Quotient Rings.
You didn't specify whether you're talking about the right socle or left, but I think it isn't important for answering the question at hand, so I pick the right hand side.
If you represent $T$ with row finite (or column finite) matrices, then each row contains a copy of $U_T$, which is a simple $T$ module and is therefore contained in $soc(T_T)$. There are infinitely many rows, though, so you can easily build an infinite descending chain (and an infinite ascending chain) of submodules of $soc(T_T)$ by "stacking" these submodules.
This can't occur in a semisimple ring because of the ACC and DCC on right ideals.