How to remove a tone/interference from a signal, without using classical linear filtering I am wondering if a methodology(ies) exist for essentially 'filtering' out a tone or interference, WITHOUT resorting to classical linear filtering techniques. (band pass filter, etc for example). 

The reason for this question is because inherent in any classical linear filtering operation, I am at the end of the day linearly combining samples among themselves, and while this might do well to remove a tone of another frequency that I am trying to remove, it also has the effect of 'smearing' my signal in time. This smearing effect causes information such as the 'sharpness' of the signal to be lost, and in this specific case, the 'sharpness' carries information such as start of signal energy, etc, that I use later. 

What method(s) exist (do they exist?) for this goal? 

I should also add, please assume omnidirectional case, (I only have one sensor), but if assuming a case with multiple sensors lends itself to a solution, feel free to use that assumption as well.

Thanks

EDIT#1:

I must say that for the tonal case,  I think it might be straight forward as many of you have indicated, vis-a-vis estimating the tonal parameters, reconstructing, and subtracting. The actual problem though, is a little more complicated... 

The unwanted signals can range from anything like a group of whales having a 'conversation' to a boat's motor overhead. In such cases no known robust model exists, all I know is that my signal of interest is among all those other energies - they are seperable in frequency, but I need a way to remove them as well, without filtering because that removes my sharpness. 

FYI, I think this falls into the realm of non-linear filtering or other techniques from different areas? I should also add that I have a lot of computing power.
A sharp edge or transient may require a very wide-band spectrum to represent it.  Thus any classical or frequency domain filter that completely removes a spectral band could unsharpen or "smear" a sharp transient, by removing a necessary component of that transient.

There are two non-linear methods based on analysis/resynthesis techniques that might work.  Either estimate your tone or interference signal precisely enough to synthesize it and subtract it.  Or characterize your desired signal well enough to resynthesize it without the noise.  You will need a very accurate model for either of these.  Both of these will only remove or add the appropriate fraction of energy in any spectral band, instead off all or none, or some random percentage (relative to the actual S/N) given by a filter response curve.