In both Arabic and Hebrew we see that the greeting encompasses a notion of peace. Let's see what one source has to say similarly about Phoenician-Punic. I am providing that which I see most relevant to your post.
Š-L-M I
[Heb. s̆-l-m]
v. pi’’el 1. GREET SOMEONE (+ 'lt)
CIS I 5510.6/7 (Pu) qr’ lmlqrt ysp ‘lty l s̆lm wlyrhy bmqm [z], “As for him who calls to Milqart, they shall continue to greet him and make him welcome in this city.” Cf. Arabic sallama 'alā.
ŠLM II
[Heb. s̆alōm]
n.m. 1. PEACE, specifically PEACEFUL RELATIONS
KAI 26 a I 11/12 (Ph) ws̆t ‘nk s̆lm ‘t kl mlk, “And I made peace with every king.”
ŠLM III
3. PROSPERITY, WELL-BEING
KAI 26 A III 1/4 (Ph) wbrk b’l krntrys̆ ‘yt ‘ztwd hym ws̆lm w’z ‘dr ‘l kl mlk, “May Baal-KRNTRYŠ bless Aztwadda with long life an [sic] well-being and might greater than that of any king.” – CIS I 5511.4 (Pu) dr s̆lm wmnh[t], “A time of prosperity and peace.”
Source: Krahmalkov, Charles R. Phoenician-Punic Dictionary. Leuven: Uitgeverij Peeters, 2000. 462-64.
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