**No**, not every matroid satisfies this property.  For example, it is known to fail for the cycle matroid of $K_4$.   The matroids that satisfy your property are called *base orderable* matroids. There are important classes of matroids that are base orderable, such as transversal matroids.  Moreover, base orderability is a minor-closed property, but Ingleton proved that there are actually an infinite number of excluded minors.  See these [slides][1]
of Joseph Bonin for more information. For example, the slides include a proof that $K_4$ is not base orderable.  

  [1]: https://blogs.gwu.edu/jbonin/files/2016/04/BaseOrderable-rjuvq2.pdf