"I couldn't produce the police clearance and my case was in administrative processing because of it. In Bangladesh, they say they usually finish administrative processing in 60 days. I explained why I couldn't produce the police clearance by letter. Is it possible to have a K1 visa without the police clearance?"
As far as getting approved - a fiance visa is highly unlikely to get approved without providing all required documents.
A police certificate is probably one of the most important documents that they're reviewing. That goes for both the US citizen sponsor and the alien beneficiary. They perform background checks on both and they want to make sure that the alien beneficiary coming to the US is not going to be a harm to the US or their citizens, so they perform these background checks.
How they do that is they request police clearance from every country that you lived or worked in since age 16. You must provide these documents. We've heard of people submitting letters before, stating why they were unable to obtain certain documents, but we've never heard of a consular officer actually accepting those. So, you may receive what is called a Request for Evidence and they'll ask you to obtain those.
Now, there are some exceptions to the rule.
That is when the embassy itself states that if you are living or residing outside of this country, you can no longer obtain said records from this particular country. To find out if this is the case in your particular country, check the State Department's Reciprocity Schedules.