exoplanet - Is there any way a planet could form independent of a star?

Well you need to see the related question brown dwarfs and planets , because the answer to your question depends on how you define a planet.

If you demand that a "planet" has a rocky core then it seems very unlikely that a planet could form in isolation away from a parent star. The parent star is needed in order to differentiate the rocky material from the gas and allow it to condense.

On the other hand, if you wish to define a planet as simply an object below a certain mass (say the deuterium burning threshold at 13 Jupiter masses) then it seems very likely that such an object could form in isolation. They would be entirely gaseous, but there would be little to distinguish them from brown dwarfs at only slightly higher masses.

At present there are plenty of candidate "free-floating planetary mass" objects.
For example see Joergens et al. (2014); Liu et al. (2013); Zapatero-Osorio et al. (2000). Unless we have our understanding of the physics completely wrong, then it is likely that at least some of these are lower than 13 Jupiter masses. However, their origin remains unclear. It is possible they could all have formed around stars and then subsequently been ejected, but the presence of significant numbers of these objects in young star forming regions and the lack of $sim$10 Jupiter-mass objects orbiting stars, suggests that there is an alternative formation scenario that can produce such objects in isolation.

What could these formation scenarios be? These low-mass objects could just be an extension to lower masses of the fragementation process that forms stars; they could be ejected embryos that started their lives in multiple systems; they could be "failed" stellar cores that could not accrete more gas because of photoevaporation by nearby massive stars; or they could form by gravitational instability around stars with unusually massive disks and be ejected by a close encounter with another star. These possibilities are reviewed by Whitworth et al. (2006) and Chabrier et al. (2014), and are all still thought plausible to some extent.