Generally, no, but in at least one notable exception, Yes.
@Dave has given a good answer that covers much of the industry. However, there is an industry sector for which the answer is, "Yes": FAA certified Special Light Sport Aircraft (S-LSA). These maintenance requirements are found in part 91, not—curiously enough—in part 43.
Firstly, the FAA gives manufacturers of S-LSA the authority to prescribe what must be done for routine maintenance and inspections. This authority is not absolute, as part 43 also applies and the role is also open to "a person acceptable to the FAA".
See 14 CFR 91.327 (b) (1) and (2):
(b) No person may operate an aircraft that has a special airworthiness
certificate in the light-sport category unless—
(1) The aircraft is maintained by a certificated repairman with a
light-sport aircraft maintenance rating, an appropriately rated
mechanic, or an appropriately rated repair station in accordance with
the applicable provisions of part 43 of this chapter and maintenance
and inspection procedures developed by the aircraft manufacturer or a
person acceptable to the FAA;
(2) A condition inspection is performed once every 12 calendar months
by a certificated repairman (light-sport aircraft) with a maintenance
rating, an appropriately rated mechanic, or an appropriately rated
repair station in accordance with inspection procedures developed by
the aircraft manufacturer or a person acceptable to the FAA;
In addition to this, manufacturers of S-LSA can issue Safety Directives, which are generally mandatory and can be thought of as roughly equivalent to Airworthiness Directives (ADs). Again, this power is not absolute.
See 14 CFR 91.327 (b) (4):
(4) The owner or operator complies with each safety directive
applicable to the aircraft that corrects an existing unsafe condition.
In lieu of complying with a safety directive an owner or operator may—
(i) Correct the unsafe condition in a manner different from that
specified in the safety directive provided the person issuing the
directive1 concurs with the action; or
(ii) Obtain an FAA waiver from the provisions of the safety directive
based on a conclusion that the safety directive was issued without
adhering to the applicable consensus standard;
1 That is, the S-LSA manufacturer.
Furthermore, any repairs or alterations must be approved by the manufacturer, either by the existing manufacturer supplied Maintenance Manual, or on a case by case basis for those falling outside the scope of the Maintenance Manual. Once again, this power is not absolute; the FAA has to retain some authority.
See 14 CFR 91.327 (b) (5) and (6):
(5) Each alteration accomplished after the aircraft's date of
manufacture meets the applicable and current consensus standard and
has been authorized by either the manufacturer or a person acceptable
to the FAA;
(6) Each major alteration to an aircraft product produced under a
consensus standard is authorized, performed and inspected in
accordance with maintenance and inspection procedures developed by the
manufacturer or a person acceptable to the FAA; and
See also this article by D. Martino: Work Books: A flight instructor’s guide
to S-LSA maintenance regulations, which gives a good interpretation of these rules with less legalese.