Yes and no.
If what you want is an assurance that some current version of PuTTY which you've already downloaded will remain free, then you already have that assurance: it's called the PuTTY Licence. It grants you permission to use, distribute and copy the software to which it applies; once we've granted that permission (which we have), we can't just revoke it.
On the other hand, if you want an assurance that future versions of PuTTY won't be closed-source, that's more difficult. We could in principle sign a document stating that we would never release a closed-source PuTTY, but that wouldn't assure you that we would keep releasing open-source PuTTYs: we would still have the option of ceasing to develop PuTTY at all, which would surely be even worse for you than making it closed-source! (And we almost certainly wouldn't want to sign a document guaranteeing that we would actually continue to do development work on PuTTY; we certainly wouldn't sign it for free. Documents like that are called contracts of employment, and are generally not signed except in return for a sizeable salary.)
If we were to stop developing PuTTY, or to decide to make all future releases closed-source, then you would still be free to copy the last open release in accordance with the current licence, and in particular you could start your own fork of the project from that release. If this happened, I confidently predict that somebody would do that, and that some kind of a free PuTTY would continue to be developed. There's already precedent for that sort of thing happening in free software. We can't guarantee that somebody other than you would do it, of course; you might have to do it yourself. But we can assure you that there would be nothing preventing anyone from continuing free development if we stopped.
(Finally, we can also confidently predict that if we made PuTTY closed-source and someone made an open-source fork, most people would switch to the latter. Therefore, it would be pretty stupid of us to try it.)