4000 dates in a dropdown is something you should avoid at all costs. There are different better ways depending on your needs and technical ability. The easier one is to simply use a filter, like this:

in the example above, the filter is broad (see the ALL options), so it generates a limited range of dates which can be navigated with a slider (I have added both together, but the slider should show after user filtering, of course) . You can use a strict filter, which won't generate any range, but specific dates, as such:
select parameter --> (display only years with active dates for teh parameter) --> select year --> (display only months with active days) --> select month --> (display only active days within selected month) --> select day
There are many other options to do this, but I think the 2 above are easy to implement and very easy to understand by users