William Ackman, the famed shareholder activist, is losing investors at a rapid pace and facing a future that would no longer include managing a private hedge fund.

After three years of subpar performance, most investors in his Pershing Square Capital Management LP asked for money back, and pulled about two-thirds of the cash that could be withdrawn at the end of last year, according to people familiar with the matter. Redemptions, which are limited to a portion of total assets, have continued at a similar rate this year, one...