Extension of Ulrich bundles

We study extension of Ulrich bundles from a smooth nondegenerate subvariety $X$ of ${\mathbb{P}}^n$. If $X$ is a complete intersection of dimension $\ge 2$, we show that the extension is not possible except in the trivial case. For an arbitrary $X$, we characterize when the extension is possible, assuming some condition on the extended vector bundle. As an application, we generalize previous results of L{ó}pez and Zamora. We also give several classes of examples of Ulrich bundles on curves that extend to the ambient projective space.