Can you make a laser without cavity, if you have a pump strong enough? I think no, but is my reasoning correct?

Masers, and even optical frequency lasers, occur in molecular clouds. These are single-pass processes rather than resonant cavities and so not always regarded as a "true" laser. See:

I think you basically answer the question yourself and @mikestone seems to make the same point: you can have strong light amplification without cavity. However:

  • This is an inefficient use of the active media
  • You will have no control over the modes, including the direction of the emission.

Whether one may call this a laser/maser or simply and amplifier is a matter of terminology.

Remark Terms laser/maser (light/microwave amplification by stimulated emission of radiation) are inherently ambiguous. This is why one would sometimes talk about quantum generators as opposed to quantum amplifiers, in analogy with the traditional generators and amplifiers, where the former are characterized by a periodic cycle (like a photon bouncing back and forth between the mirrors).

Congratulations, you've invented one widely-used type of :P Wikipedia has quite a bit of stuff about uses for stimulated-emission without a cavity, including amplifying the output of an existing laser (e.g. as a laser guide star as a reference for adaptive optics in a telescope).

Apparently some designs (like a semiconductor optical amplifier) that are naturally built a lot like a laser diode would be have to intentionally avoid reflections to not create a cavity. Even so, amplified spontaneous emission is still a source of noise when your goal is to amplify existing signals. Non-linearity of semiconductor amps can apparently be used for optical signal processing.

It's not in general limited to amplifying a single frequency, though. Real life repeaters for long-haul fiber-optic data links use this to amplify all the light of different wavelengths in a fiber carrying different signals on different wavelengths, e.g. 1310 nm and 1550 nm or a range of wavelengths around those. (WDM = Wavelength-Division Multiplexing; like using a prism to get signals from different modulated lasers in, and another prism (or actually etalons) on the receiving end to split it up for different detectors.)

Optical repeater amplifiers are used to boost signal strength in long-haul fiber optics data links (under-sea / between cities). Every few hops you need to clean up the signal (because of modal dispersion even in single-mode long-haul fiber and other effects that cause a pulse to spread out in time). So every few hops you use a more expensive repeater that digitizes the incoming signals (on each wavelength) and then modulates an output laser. Or maybe something clever that doesn't go all the way back to electrical.

Wikipedia's Optical Amplifier article starts off:

... An optical amplifier may be thought of as a laser without an optical cavity, or one in which feedback from the cavity is suppressed. Optical amplifiers are important in optical communication and laser physics. They are used as optical repeaters in the long distance fiberoptic cables which carry much of the world's telecommunication links.

There are several different physical mechanisms that can be used to amplify a light signal, which correspond to the major types of optical amplifiers. In doped fiber amplifiers and bulk lasers, stimulated emission in the amplifier's gain medium causes amplification of incoming light. ...

(This doesn't imply that the only thing you can do with a non-cavity laser-like thing is amplify optical signals. But if you do want to amplify optical signals, stimulated emission is your best bet.)