Deputy of Detention’s capacity would nonetheless resolve and performance usually. You must, amongst others, destroy or exile the Deputy to forestall the impact from exiling your permanents.

Deputy of Detention’s capacity is a triggered capacity that triggers when the Deputy enters the battlefield. You cannot have an effect on the Deputy with the Trickster earlier than the Deputy has entered the battlefield, and afterwards, it is too late. Once the flexibility has triggered, the flexibility that’s created on the stack capabilities independently of its supply.

113.7a Once activated or triggered, a capability exists on the stack independently of its supply. [..]

Even if you happen to take away the Deputy’s skills by way of Merfolk Trickster in response to the set off, it doesnt matter. You’d simply take away the flexibility after it has already performed its job, which is to create a capability on the stack. That capacity on the stack will nonetheless resolve and exile its targets till the Deputy that created it leaves the battlefield.

You can, nonetheless, stop your permanents from being exiled. The commonest manner is to take away the Deputy from the battlefield earlier than its capacity resolves, similar to by destroying or exiling the Deputy. In that case, the flexibility nonetheless resolves (it is impartial of the Deputy, keep in mind) but it surely will not do something to your permanents:

610.3. Some one-shot results trigger an object to vary zones “until” a specified occasion happens. A second one-shot impact is created instantly after the desired occasion. This second one-shot impact returns the item to its earlier zone.

Deputy of Detention’s capacity is such a one-shot impact.

610.3b If a resolving triggered capacity creates the preliminary one-shot impact that causes the item to vary zones, and the desired occasion has already occurred earlier than that one-shot impact would happen however after that capacity triggered, the item doesn’t transfer.

The Deputy says that the focused permanents are exiled “till Deputy of Detention leaves the battlefield”, and 610.3b says that if the Deputy has already left the battlefield earlier than its capacity has exiled its targets, nothing is exiled in any respect.