Theoretically: Yes you can and yes it's free. But in the end, you'll end up with a lot of work and maybe even issues. Magento 2.x is not just an upgrade from Magento 1.x, it's actually an entirely new platform. There are tools to migrate, but I'm not sure if they'll give you the result you want, I don't think it's an out-of-the-box solution.
But within Magento, you'll have your templates and extensions. These are not compatible with Magento 2, so you should get the Magento 2 version of the extension and redo the templates.
So, is upgrading to Magento 2 wrong? The answer to that is: No. Magento 1 will be unsupported somewhere in 2018 (3 years after releasing Magento 2), so there won't be any security fixes or other updated anymore. So in the end, when you want to keep using Magento, you should upgrade to Magento 2. And you can do this yourself or you can let a Magento agency do it for you. In that case: moving to Magento 2 won't be free anymore.
I hope this answers your question. If not, just let me know.
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