Do countries with centrally-planned economy have Entrepreneurship as a factor of production? Let's don't count self-employed people without other employees (like &quot;lone wolf&quot; craftsmen) and people who have an illegal business as entrepreneurs. By centrally-planed economy I mean an economy without legal private sector (or where its contribution to GDP is so small, that it can safely be ignored).
If the answer is &quot;yes&quot;, then how? For an example, can we count a state-owned factory director as an entrepreneur?
This is hard to answer because it is disputable if entrepreneurship is a factor of production just to begin with. There are economists who would argue that any entrepreneurial activity already falls under labor/human capital.
Setting the caveat above aside there are two major views on how entrepreneurship as a factor should be treated (see here). In one view entrepreneurship cannot be divorced from ownership of assets since as argument goes people can take entrepreneurial risk only if they actually have 'skin in the game'. In centrally planned economy asset are collectively owned state and all market risks are socialized. Under this view there can be no entrepreneurship in centrally planed economy. Moreover, managers would not count under such view either.
A second view would treat entrepreneurship as purely an intellectual activity of combining different factors of production (to be clear the previous view also agrees on this part but they add assets as additional requirement). For example, in Fed St. Louis series for students the entrepreneurship as a factor of production is defined as:

The fourth factor of production is entrepreneurship. An entrepreneur is a person who combines the other factors of production - land, labor, and capital - to earn a profit.

The above definition does not have asset ownership as a part of requirement and manager's work here could be considered to be an entrepreneurial factor. The additional important requirement is that this is done to earn profit but as far as I understand here it does not matter if the payment is considered profit or wage in accounting manner but rather from economic perspective. Furthermore, generally speaking profits can exist in centrally planned economies, to the extent these are shared with the managers in some explicit or implicit way, you could argue that under this definition entrepreneurship can exist in centrally planed economy.
This being said, centrally planned economies most likely don't have much of it. According to highly cited work of Audretsch (2007):

Measurement of entrepreneurship capital is no less complicated than is measuring
the traditional factors of production. Just as measurement of physical capital, labour,
and knowledge invokes numerous assumptions and simplifications, creating a metric for
entrepreneurship capital has also presented a challenge. Many of the elements that constitute
entrepreneurship capital defy quantification. In any case, entrepreneurship capital, like all
of the other types of capital, is multifaceted and heterogeneous. However, entrepreneurship
capital manifests itself in a singular way—the start-up of new enterprises.

The emphasis was added by me. Hence an entrepreneurship can be measured by the rate at which new enterprises form (and in fact this is the way in which most literature measures it). It goes without saying that historically speaking centrally planned economies are not exactly known for high rates of new enterprise formation.