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

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