Is it permissible to give an interestless loan in which the loaning party holds a sentimental item of the one taking the loan until the loan is paid?
رهن is permissible as proven from the Quran and Sunnah:
وإن كنتم على سفر ولم تجدوا كاتبا فرهان مقبوضة فإن أمن بعضكم بعضا فليؤد الذي اؤتمن أمانته وليتق الله ربه
And if you are on a journey and cannot find a scribe, then a security deposit [should be] taken. And if one of you entrusts another, then let him who is entrusted discharge his trust [faithfully] and let him fear Allah, his Lord.
اشترى رسول الله صلى الله عليه وسلم من يهودي طعاما ورهنه درعا من حديد
Allah's Messenger (ﷺ) bought from a Jew grain (as loan) and pledged him his iron coat-of-mail.