$\operatorname{Ext}(\mathbb{Z}_{p^\infty}, \mathbb{Z}_p)$

Choose generators \(x_n\) of \(\mathbb Z_{p^\infty}\) with \(x_n\) of order \(p^n\) and \[ p x_{n+1}=x_n. \] Let \(F_0\) be free on generators \(e_1,e_2,\dots\), mapping \(e_n\mapsto x_n\). The kernel is free on the relations \[ r_0=p e_1, \qquad r_n=p e_{n+1}-e_n\quad(n\ge1). \] Thus we have a free resolution \[ 0\longrightarrow F_1\xrightarrow d F_0 \longrightarrow\mathbb Z_{p^\infty}\longrightarrow0, \] where \(F_1\) is free on \(r_0,r_1,\dots\) and \(d\) is given by the displayed formulas.

Apply \(\operatorname{Hom}(-,\mathbb Z_p)\). Since Hom out of a direct sum is a product, \[ \operatorname{Hom}(F_0,\mathbb Z_p)\cong\prod_{n\ge1}\mathbb Z_p, \qquad \operatorname{Hom}(F_1,\mathbb Z_p)\cong\prod_{n\ge0}\mathbb Z_p. \] If a homomorphism \(F_0\to\mathbb Z_p\) is represented by \((a_1,a_2,\dots)\), then \(d^*\) sends it to \[ (0,-a_1,-a_2,-a_3,\dots), \] because multiplication by \(p\) is zero in \(\mathbb Z_p\). Therefore \[ \operatorname{im}d^* =\{(b_0,b_1,b_2,\dots):b_0=0\}, \] and hence \[ \operatorname{coker}d^*\cong\mathbb Z_p \] by projection onto the \(r_0\) coordinate. By the definition of Ext from this free resolution, \[ \boxed{\operatorname{Ext}(\mathbb Z_{p^\infty},\mathbb Z_p)\cong\mathbb Z_p.} \]