I've come across this sentence with ~すまいて after a verb. I've never seen it like that before, but I'm assuming it is a form of ~済まい:
「いくらも進んでおりませぬよ。儂の命のあるうちに終わりますまいて」
I can't make any progress. It probably won't end during my lifetime.
If it is ~済まい, what does the て do here...?
まい is a somewhat old word with an approximate meaning similar to ～ないだろう (here). You may or may not have heard the phrase ～じゃあるまいし. It attaches straight after the dictionary form of godan verbs, and replaces ～る for ichidan verbs; like with godan verbs, it goes straight after ～ます in polite speech.
This ～て is a sentence ending particle that acts for emphasis, like よ or ぞ. It's stereotypically associated with older speakers, which, judging by the use of 儂 and おりませぬ, the speaker is.
And speaking of おりませぬ, it's an old form of おりません, the polite negative of おる, which is a humble or dialectal form of いる - here used out of an association with older speakers. It is not potential. The excerpt given says:
I have made no progress/I am making no progress. Most likely it shall never end within my lifetime.
Oh, and in more normal Japanese:
いくらも進んでいませんよ。私の命のあるうちに終わらないでしょう。
Angelos explained well. I just wanted to add some references for て (kindly provided by reddit).
明鏡国語辞典
助動詞「だ」「じゃ」「ぬ」「まい」「だろう」などに付いて》自分や相手に言い聞かせる気持ちを表す。「大変なことじゃて」「そんなことはあるまいて」
新選国語辞典 第十版
〘古語〙軽く相手に説き聞かせる。「ゆだんがならぬ━」
大辞林 第四版
動詞・助動詞の終止形に接続する。近世以降の用法。みずからうなずく気持ちで軽く添える。「油断がならぬ━」「おれが行くこともあるまい━」「女の子は意地の悪いものでございます━／滑稽本・浮世風呂（二）」
