Keeping job application secret Is it reasonable to request keeping my application to a tenure-track position secret in the cover letter?
Would search committee likely to respect such a request?
If at all possible, how long can such secret be kept?


The motivation for keeping an application secret is to not create any problem with the current department.
There are no guarantees, but most places will respect your wishes for a while. But they may want to contact your references, for example. 

They might also have to make a public statement if you are offered a job, depending on various rules. 

A complicating factor is whether every committee member gets the word. And if they explore your candidacy with other colleagues (not always allowed) your wishes might not get communicated or respected by everyone. So, you assume some risk, but it is likely small. 

And your situation isn't unique in any way. Lots of people need to do this, so most committee members are probably aware of it and have seen it before.