What you are asking for is not all that uncommon.
The recommendation is to purchase 20m-30m of 8mm cord/rope. An example of that is the Petzl Conga which is semi-static and mfg approved for handlines. Old climbing rope can be used in the same application, the dynamic element does not make it unsafe, provided it is reasonable in good shape and has not seen more than its rated number of falls. Recommendation is to cut the rope to 20-30m if you are carrying climbing rope.
Any rope system has to be examined with respect to the forces applied.I
For example. If you are only going to be anchoring via yourself, then there is no need for any strength in your rope, you won't be able to hold much more than a few hundred pounds of static weight without slipping off your stance. (#1 - Pulling someone out of a swamp on moorland without getting yourself into trouble too )
If you carry only a rope with no additional gear you will be limited to natural anchors like trees and stones, and your ability to belay or abseil limited by the friction of your body or hands. (#2 - Abseiling someone down)
Once you start putting more force on the rope, either with dynamic actions, higher angle, or more weight, the basic system and body friction technology will not be sufficient to be safe. So components of the system will need to be upgraded, harnesses, belay devices, slings, and carabiners. To amplify the friction and holding power. The 8mm rope can remain as part of that system, provided the angle remains low (#3 - Pulling someone from a ditch)
For low angle terrain, handlines, or confidence ropes can provide a handfold and make travel convenient and quicker, while remaining safe. But in steeper terrain, or where a fall can result in serious injury/death - a handline may present a false sense of security and the 8mm size will not be safe without additional equipment. At the very least the recommendation is to carry a harness and set of prusik slings. The prusik slings are a natural dynamic load limiter and will slip at forces that are below that which will break the rope. (#4 - Confidence rope whilst hiking )
Application of mechanical amplification via pulley systems can greatly increase the tension on the rope beyond anything that can occur in belay or rappel situation. It is easy to generate forces that can break even rescue rated 13mm static rope let alone climbing rope or 8mm cord. In the field mechanical amplification is used to effect rescue by helping to raise a subject, such as out of a crevasse, and this can be done safely with climbing rope (not 8mm line) . But in the case of vehicle winching, the amount of force can easily exceed the breaking point of the rope, so towing equipment which is typically 4-5" wide flat webbing and some dynamic elements is carried instead. (#5 - Winching a car out of trouble in winter weather conditions using a pulley system)