No, unfortunately that's a weed. Modern varieties have been bred to put out fewer runners, because it takes energy away from fruiting. But yes, if you get a runner, you can propagate a new plant that way. Later in the season you might get some.

A strawberry runner will look just like a smaller version of the strawberry plant, on a fleshy stem.

Strawberry runners- they will be attached to the mother plant not growing separately out of the soil.

Some strawberry plants types don't have runners.