Increasing the apparent distance between camera and subject by using mirrors, as described in the question, cannot be done reasonably within a lens. However, mirrors can be used to increase apparent distances in limited space if a permanent installation is acceptable.

Flat Mirrors

With a single flat mirror, if subject and camera are the same physical distance from the mirror, the mirror will need to be half the size of the subject plus whatever background will be captured because the angle of reflection is equal to the incidence angle. This will double the apparent distance between camera and subject.

The size of the mirror can be decreased if it is moved closer to the camera than it is to the subject. But by doing so, the apparent distance between camera and subject will approach the actual distance, which would defeat the purpose of using mirrors in the first place.

Curved Mirrors

All lenses create and correct distortion, so any distortions created by curving mirrors are reversible. Curving the mirror can be used to decrease the physical distances while retaining the apparent distances. However, it does so at the expense of increasing mirror surface area.

As with flat mirrors, the farther the subject is from the mirror, the close the apparent distance will be to the actual distance, reducing the effectiveness of the contraption.

In Practice

A lot of mirror surface area would be needed. The ratio of distances won't work out to significantly alter perspective much better than taking a few steps back.

If a permanent installation is acceptable, what you'd end up doing is putting a large mirror on a wall. Then you would have to avoid capturing the camera in the frame, as well as keep the mirror clean. This is how some optometry/ophthalmology offices are set up to increase the "distance" between patients and eye charts in small rooms.

Decreasing Distance

Fiber optics has been used to do the opposite of what you ask to improve and save lives.