〜てやまない
never cease to, sincerely (hope, love)
Verb て-form (願う, 祈る, 愛する, 期待する) + やまない / やみません
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てやまない expresses a deep, lasting feeling that never stops: 平和を願ってやまない, "I never stop hoping for peace".
It goes with a small set of verbs of hoping, loving and respecting: 願う, 祈る, 愛する, 期待する, 尊敬する. It's formal and heartfelt, and it's a staple of speeches, letters and closing greetings: 皆様のご健康を願ってやみません, "I sincerely wish you all good health".
It can describe others in the past: 母が愛してやまなかった町, "the town my mother loved so dearly".
It's literally "doesn't stop (feeling)". Compare てたまらない (N3), "so … I can't stand it", which is casual and about intense physical or emotional states.
Examples
皆様のご健康を願ってやみません。
I sincerely wish you all good health.
世界の平和を願ってやまない。
I never stop hoping for world peace.
彼は、母が愛してやまなかった町に帰った。
He returned to the town his mother had loved so dearly.
皆様の今後のご活躍を期待してやみません。
I look forward to your continued success.
子どもたちの幸せを祈ってやまない。
I never stop praying for the children's happiness.