The singular values do have a sound geometrical meaning. The first one is nothing but the operator norm of $T$, that is the maximum dilation coefficient $$\frac{\|Tx\|}{\|x\|}.$$ The next ones can be seen also as maximum of dilation coefficients, provided you replace lines by subspaces. For instance $$s_k=\max_{\dim F=k}\min\left\{\frac{\|Tx\|}{\|x\|};x\in F,x\ne0\right\}.$$ In terms of the exterior algebra over $\mathbb C^k$ ($k=m$ or $n$), you have $$s_1\cdots s_k=\sup\left\{\frac{\|Tx_1\wedge\cdots\wedge Tx_k\|}{\|x_1\wedge\cdots\wedge x_k\|};x_1,\ldots,x_k\in F\right\}.$$
Notice also that there is a $p$-adic version of the SVD decomposition. See K. S. Kedlaya. $p$-adic differential equations. Cambridge Studies in Advanced Mathematics, 125. Cambridge University Press, Cambridge, 2010
Edit. I should have also given the formula $$s_1+\cdots+s_k=\max\{{\rm Tr}(PTQ); P\hbox{ and } Q \hbox{ are unitary projectors of rank } k\}.$$ This has the interesting consequence that $T\mapsto s_1+\cdots+s_k$ is a convex function.
Edit (bis). Actually, $T\mapsto s_1\cdots s_k$ is rank-one convex. This means that it is convex along every line $A+{\mathbb R} B$ for which $B-A$ is a rank-one matrix.
