As post-doc you are typically an employee of some research institution (which is often also the case when having "your own money"). This poses some duties on you. E.g., you typically cannot work at a competing institution (whether industry or public research) unless your employer allows this. Where I am (Germany), there are also limits on working hours per day/week that limit how many hours you could work in industry in parallel to your usual employment contract - even if the business is completely different from your employer's (in which case your employer must approve of the additional job/internship).
The consequence of this is that you need to talk to your institution about the plan and get approval for this.
If your funding is in the form of a scholarship without an intermediate employer, the terms likely require you to get approval from your funding agency.
The details (e.g. whether you need to take time off) are then negotiated between you and your employer or funding agency.
I.e., you negotiate an additional contract that says you can do this internship.
As one example of a context where this is actually encouraged, and done during working hours:
EU COST Actions fund short-term scientific missions where someone (e.g. a post-doc) from one institution visits another institution, e.g. an industry partner.
(Must meet some other criteria, though. E.g., being cross-borders and being in line with the scientific goals of the COST Action)