The most natural (and neutral) way would probably be 走り続ける (はしりつづける). 走る is the verb for running. 駆ける sounds more like you're chasing after something or have a particular goal in mind.
When you combine a verb to ~続ける you use the "stem" or "pre-masu" form of the verb.
So...
走ります
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走り
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走り続ける