The answer to your question is "yes". Such complexes were first exhibited by Hachimori in his PhD thesis. See the clear explanation on his webpage:
The $f$-vector of this complex is (1,7,19,13). I suspect that 7 vertices is minimal.
Adiprasito, Benedetti and Lutz have meanwhile (since you first asked the question) extended an idea similar to Hachimori's to arbitrary dimension in their paper:
Adiprasito, Karim A.; Benedetti, Bruno; Lutz, Frank H., Extremal examples of collapsible complexes and random discrete Morse theory, Discrete Comput. Geom. 57, No. 4, 824-853 (2017). ZBL1365.05305.
Complexes having the property of Hachimori's have arisen recently as a tool. They were used as one of the main building blocks to show that the SHELLABILITY problem is NP-complete in this paper of Goaoc, Paták, Patáková, Tancer, and Wagner. (The paper appeared in a conference proceeding, though it doesn't seem to yet be on zbMath.)
I'll mention parenthetically that Goaoc, Paták, Patáková, Tancer, and Wagner require also some other building blocks of the same flavor as Hachimori's complex. Their constructions for these are a bit complicated for my taste, and they don't include all of the details in the above-linked paper. My PhD student Andrés Santamariá Galvis has some simpler constructions that are based on Hachimori's example.