Let \(\|{\cdot}\| \) be a given unitary invariant norm on rectangular \(m\)-by-\(n\) matrices, and let \(A\circ B\) be the Schur (=\,entrywise) product of matrices. The author classifies \(\|{\cdot}\| \)-isometries \(\Phi:M_{m\times n}\to M_{m\times n}\) which are Schur multiplicative, that is, which satisfy \(\| \Phi(A)\| =\| A\| \) and \(\Phi(A\circ B)=\Phi(A)\circ\Phi(B)\). When \(\| \cdot\| \) is not a scalar multiple of the Frobenius norm then such maps merely permute rows/columns of a matrix, modulo transposition and entrywise conjugation. A Frobenius norm is a special case, and there are more possibilities. The proof relies heavily on Lemmas 2.2--2.3, which are of independent interest. They distinguish scalar multiples of the Frobenius norm from other unitary invariant norms. To give a glimpse: a unitary invariant norm \(\| \cdot\| \) on complex matrices is a multiple of the Frobenius norm precisely when, for each diagonal \(D\in M_{m-2\times n-2}\), \(\| A(e^{i\varphi})\oplus D\| \) is constant for \(\varphi\in[-\pi,\pi]\), where \[ A(z):=\begin{pmatrix} z&1\\ 1&1\end{pmatrix}. \] A similar result is proven for real matrices.