In order to avoid detection by officers that already strongly suspect the character and really want to find those papers, the briefcase lining would have to both look and feel intact. Also, the character would probably only be able to hide a few papers that way without changing the briefcase's weight too much or making the papers' motion detectable when it's shaken- maybe in the ballpark of ten to twenty standard photocopier sheets. Might be doable if the character has the time, proper tools, and skill available to them.
"Jack, you have debauched my sloth."
Also, if the papers are in any way related to national security, the police will probably just rip the lining out, whether they can detect anything or not. Better to settle a citizen complaint afterward than let something that valuable get away.
So, a lot depends on what they suspect. If they're thinking 'spy,' then yeah, they probably ought to search everything very closely, rip out lining of briefcases and jackets, cavity search, x-rays, the whole nine yards.
If they're thinking 'drug dealer' then, so long as the lining isn't big enough to contain piles of cash, or drugs, they probably don't care.
If they're thinking 'drunk businessman' they may not bother to do more than open the briefcase.
Which is actually a fun writing point, because (assuming the character is meant to be competent) there's an opportunity here for a bit of competence porn as he tries to mislead them, not as to his guilt, but as to the cause of it.
That sounds like it would be fun to write.
I have a scene where a character is sneaking some sensitive papers to a secret spot and is caught by the police and taken to jail for questioning. Now they hide the papers in the lining of their briefcase. The police search the briefcase, but find nothing. However, thinking about it some more, is there a reason why the police wouldn't search the lining of the briefcase or notice that something was stuffed in the lining in the briefcase?