I think you want Theorem 2.1 On the Ehrhart Polynomial of Minimal Matroids by Ferroni. Note Theorem 2.1 cites previous work that may be of interest, but I know of these matroids from this recent paper.

Theorem 2.1 says if you have a connected matroid on $n$ elements of rank $k$ then the number of bases is at least $k(n-k)+1$. Furthermore, there is a unique matroid up to isomorphism that realizes this bound (and that matroid turns out to be graphic). The construction of this matroid is described in Section 2 of the linked paper.

I think you want Theorem 2.1 On the Ehrhart Polynomial of Minimal Matroids by Ferroni. Note Theorem 2.1 cites previous work that may be of interest, but I know of these matroids from this recent paper.

Theorem 2.1 says if you have a connected matroid on $n$ elements of rank $k$ then the number of bases is at least $k(n-k)+1$. Furthermore, there is a unique matroid up to isomorphism that realizes this bound (and that matroid turns out to be graphic). The construction of this matroid is described in Section 2 of the linked paper.

Edit: It has been clarified that the question wishes to address minimality by inclusion of the set of bases. The answer above completely deals with the minimal by cardinality case (which is the subcase of the whole question). In Remark 2.9 of the paper linked an example of a minimal by inclusion matroid which is not minimal by cardinality is given. So, there are indeed other matroids OP is curious about.