There are a lot of different strawberry plants nowadays, some are designed to tumble, but I would say 'yes'.
The other possibility is to bury the tip in a pot of soil. They will quickly root and then you can cut them off the main plant and have new plants. With new plants if you take all the flowers off the first year you get so many more fruits the second year it more than makes up for it. When I moved and brought rooted runners with me I made sure there were enough of them I only took flowers off half and still had some fruit next year.
The strawberry we know as 'normal' is a hybrid between one that grows in costal scrub in S America and one that grows in open woodland in Pennsylvania, but there are literally hundreds of native species scattered across the globe and people have been experimenting with all sorts of crosses recently. That original cross turned up in France in the sixteen hundreds and the Dauphin sent King James one, but he still ordered a couple of hundred runners of native English strawberries to be rooted. The fruit is tiny by comparison, and a lot of work to produce and pick, but he preferred the flavour, and Kings don't do the work themselves.