Praise be to Allah.
In Wikipedia, it says: The salamander resembles the wazagh [a type of lizard]. Most of them live in water throughout their lives; some of them move to water periodically, whilst others are completely land-based in their adult form.
The pictures in the Wikipedia article show that salamanders do not come under the heading of the wazagh lizard that we are enjoined to kill. Not everything that resembles the wazagh, such as other types of lizards (Lacertilia), is regarded as being the same as it or as coming under the same ruling. The wazagh is harmful and has an evil nature, unlike many of the other lizards that resemble it.
It says in al-Mausuu‘ah al-‘Arabiyyah al-‘Aalamiyyah: The salamander, which resembles a lizard, is a weak animal and is not harmful. End quote.
If the salamander is not a type of wazagh in terms of its physical makeup, and does not resemble it in terms of having a harmful and evil nature, then it does not come under the same ruling.
Ad-Dumayri said regarding the giant girdled lizard (cordylus giganteus), which is a lizard that resembles the wazagh:
It is a small, smooth-skinned animal that runs around a great deal. It resembles the wazagh but is not as bad as it, and does not cause harm. It is of many types, including white, red, yellow and green; all of them have black spots. The colours depend on where they live. Some of them live in sandy regions, and others live near water and vegetation; some of them are used to people and live near them.
End quote from Hayaat al-Haywaan (2/167).
At-Toorishti [a Hanafi scholar] said:
The wazagh is a small animal, and it was said that its name reflects the fact that it is light and moves quickly.
End quote from al-Muyassar fi Sharh Masaabeeh as-Sunnah (3/944).
Hence those that live in the sand – although they resemble the wazagh – are not like it, therefore it is not prescribed in the hadith to kill them. They existed at the time of the Prophet (blessings and peace of Allah be upon him) and there is no report to suggest that he enjoined killing them. Therefore a number of scholars decided that it is mustahabb (recommended) to leave them alone, because they are not harmful.
Ad-Dumayri [a Shaafa‘i scholar] said:
Our companions said: With regard to creatures that are neither harmful nor beneficial – such as dung beetles, worms, crabs, small birds, Egyptian vultures, giant girdled lizards, turtles, flies and the like – it is makrooh (disliked) to kill them, for the pilgrim in ihraam and others.
This was stated definitively by the majority [i.e., the majority of Shaafa‘is].
It is proven in Saheeh Muslim, in a report from Shaddaad ibn ‘Aws (may Allah be pleased with him) that the Prophet (blessings and peace of Allah be upon him) said: “Allah, may He be exalted, has decreed ihsaan (proficiency; kindness) in all things, so when you kill, kill well.” It is not ihsaan to kill anything for fun or for no purpose.
End quote from Hayaat al-Haywaan (1/430).
Perhaps the questioner is not aware of the fact that the Prophet (blessings and peace of Allah be upon him) said of the wazagh [a type of lizard] that it used to blow on (the fire of) Ibraaheem (peace be upon him).
It is proven in Saheeh al-Bukhaari (3359) from Umm Shareek (may Allah be pleased with her) that the Messenger of Allah (blessings and peace of Allah be upon him) enjoined killing the wazagh and said: “It used to blow on (the fire of) Ibraaheem (peace be upon him).”
Abu ‘Umar ibn ‘Abd al-Barr said: The scholars are unanimously agreed that it is permissible to kill mice and rats both outside and within the Haram zone, and to kill scorpions and wazagh.
End quote from al-Istidhkaar (4/156).
Abu Bakr ibn al-‘Arabi (may Allah have mercy on him) said:
… (Firstly): Animals are of two types: harmful and not harmful.
Those that are harmful are to be killed, and those that are not harmful are not to be killed.
The wazagh is harmful by nature because it blew on (the fire of) the Prophet of Allah, which indicates that it is innately harmful.
It causes damage to food, by contaminating it and spoiling it, and it may kill the one who eats that spoiled food. So it is obligatory to kill it and to kill any other creature that is like it.
(Secondly): if an animal is not harmful, it is not permissible to kill it.
End quote from ‘Aaridah al-Ahwadhi (6/276).
Shaykh Ibn ‘Uthaymeen was asked: The Messenger (blessings and peace of Allah be upon him) encouraged us to kill the wazagh [a type of lizard] and said that “the one who kills it with the first blow will have one hundred hasanaat [rewards for good deeds].” What is the ruling on killing the wazagh and what is the ruling on killing frogs?
He (may Allah have mercy on him) replied: With regard to killing the wazagh, it is Sunnah and there is great reward in that. The Messenger of Allah (blessings and peace of Allah be upon him) said: “It used to blow the fire onto Ibraaheem when he was thrown into it.”
Moreover, it is harmful and its voice is ugly and unpleasant.
With regard to frogs, they are not to be killed, unless they are harmful; if they are harmful, then there is nothing wrong with killing them, because everything that is harmful may be killed, as the fuqaha’ (may Allah have mercy on them) said it is prescribed in the Sunnah to kill every harmful creature.
End quote from Fataawa Noor ‘ala ad-Darb,
If this little creature [the salamander] – which is not the same as the wazagh – is proven to be harmful, then it may be killed, not because it is a kind of wazagh, but so as to put an end to its harm. If it not proven to be harmful, then it is makrooh to kill it, as is the case with all other animals that are not harmful.
See also the answer to question no. 173270.
And Allah knows best.