Is it OK to apply for an advertised postdoc position a few days (not more than one week) after the deadline? It is not an on-line application form, but sending an email with attachments. Will this application be dismissed? Or more like "Let's see this candidate too". How does such hiring work? Are there any downsides (untrustworthy candidate etc.)?
The obvious question in the mind of the potential employer will be: why were you not able to meet the deadline? I would suggest you have a good reason for this. If it's just that you were disorganised, this will not reflect well on you. While you could take the attitude of "I've got nothing to lose by trying - they can only say no", bear in mind that many fields of academia are quite small communities and you may well end up wanting to apply for another job in the same lab, or collaborate in the future. Therefore, making a good impression is important even outside of trying to get the job. You need to weigh this up against how much you want that particular postdoc.
On the other hand, if you have a good reason for missing the deadline, then you might as well get in touch and see what they say. Sometimes people don't receive enough good quality applications and are delighted to include an extra candidate.
I would agree with @VonBeche's recommendation to call beforehand to ask whether they could accept a late application, before sending the completed forms.