It is quite possible. Bacteria do all sorts of weird and wonderful things to have high genetic diffusion. One of the most interesting is that bacteria will sometimes link up to other bacteria to upload a copy of their genetic codes. (!)
There is a little bit of hope that it could last for a reasonable amount of time, if properly handled. The issue is that organisms can't adapt if the dosage is too high. Usually adapting to a new threat requires, say, a dozen little changes which each improve survival by 5-10%, rather than one monolithic change which improves survival to 100%. Those combinatorics can only work out if most of the bacteria are being destroyed, not if all of them are.
Antibiotics are used this way today. Many people cancel an antibiotic regimen when they feel better, and it's a medical disaster when they do: the antibiotics need to be continued to the point of Total Destruction of all bacteria in your body, which will make you feel weaker, simply so that none of them are naturally selected to have one or two of these 5%-resistance genes.
The same thing happened with DDT and insects. There was a use of DDT -- spraying inside your house -- which had low environmental impact and which insects could not develop resistance to, because the concentrations stay high. This made it amazing as an antimalarial: mosquitoes just always perch on walls and wait until the night comes before they attack, and we could use this to always give them a lethal dosage.
Unfortunately, someone gave DDT to the farmers and then two things happened: (1) DDT got into the food supply for birds, posing serious environmental risks; and (2) insects could now receive a "partial dose" of DDT and develop those sorts of immunities, so now there are DDT-resistant insect populations.