The answer (maybe)<sup>1</sup> is no. In the following paper,

<cite authors=" Gabriel Robins and Jeffrey S. Salowe">_Gabriel Robins and Jeffrey S. Salowe_, [**On the maximum degree of minimum spanning trees.**](http://dx.doi.org/10.1145/177424.177978) Proceedings of the tenth annual symposium on Computational Geometry (SCG '94), 250-258 (1994). [PDF](https://www.cs.virginia.edu/~robins/papers/cg94_camera3_final.pdf).</cite>

> The authors proved that  under the $L_p$ norm, the maximum vertex degree over all MSTs is equal to the kissing number of the corresponding unit ball; 


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<sup>1</sup> I said 'maybe', since I have not checked the proof of the mentioned paper.